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

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

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

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.

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(9) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(10) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(11) Obligation:

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

TAKE(active(X1), X2) → TAKE(X1, X2)

The TRS R consists of the following rules:

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

(12) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(13) 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.

(14) PisEmptyProof (EQUIVALENT transformation)

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

(15) TRUE

(16) 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.

(17) 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)
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(x4)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(18) 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)
U931(X1, X2, X3, active(X4)) → U931(X1, X2, X3, X4)

The TRS R consists of the following rules:

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

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

(19) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U931(X1, X2, X3, active(X4)) → U931(X1, X2, X3, X4)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U931(x1, x2, x3, x4)  =  x4
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(20) 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.

(21) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U931(X1, X2, mark(X3), 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(x3)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(22) 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(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.

(23) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U931(X1, X2, active(X3), X4) → U931(X1, X2, X3, X4)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U931(x1, x2, x3, x4)  =  U931(x3, x4)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
active1 > U93^12


The following usable rules [FROCOS05] were oriented: none

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

The TRS R consists of the following rules:

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

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

(25) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U931(X1, active(X2), X3, X4) → U931(X1, X2, X3, X4)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U931(x1, x2, x3, x4)  =  U931(x2, x3, x4)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
active1 > U93^13


The following usable rules [FROCOS05] were oriented: none

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

The TRS R consists of the following rules:

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

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

(27) 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)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U931(x1, x2, x3, x4)  =  U931(x2, x3, x4)
mark(x1)  =  mark(x1)
active(x1)  =  active

Lexicographic Path Order [LPO].
Precedence:
mark1 > U93^13
active > U93^13


The following usable rules [FROCOS05] were oriented: none

(28) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(29) 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)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U931(x1, x2, x3, x4)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(30) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(31) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U931(active(X1), X2, X3, X4) → U931(X1, X2, X3, X4)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U931(x1, x2, x3, x4)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(32) 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.

(33) PisEmptyProof (EQUIVALENT transformation)

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

(34) TRUE

(35) 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.

(36) 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)
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)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(37) 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(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.

(38) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U921(X1, X2, X3, active(X4)) → U921(X1, X2, X3, X4)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U921(x1, x2, x3, x4)  =  x4
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(39) 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(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)

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.

(40) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U921(X1, X2, mark(X3), 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

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(41) Obligation:

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

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

(42) 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(x3, x4)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
active1 > U92^12


The following usable rules [FROCOS05] were oriented: none

(43) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(44) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U921(X1, active(X2), X3, X4) → U921(X1, X2, X3, X4)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U921(x1, x2, x3, x4)  =  U921(x2, x3, x4)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
active1 > U92^13


The following usable rules [FROCOS05] were oriented: none

(45) 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(active(X1), 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.

(46) 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, x4)
mark(x1)  =  mark(x1)
active(x1)  =  active

Lexicographic Path Order [LPO].
Precedence:
mark1 > U92^13
active > U92^13


The following usable rules [FROCOS05] were oriented: none

(47) Obligation:

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

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

(48) 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)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U921(x1, x2, x3, x4)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(49) Obligation:

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

U921(active(X1), 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.

(50) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(51) 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.

(52) PisEmptyProof (EQUIVALENT transformation)

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

(53) TRUE

(54) 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.

(55) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U911(X1, X2, X3, mark(X4)) → U911(X1, X2, X3, X4)
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)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(56) Obligation:

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

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

(57) 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)  =  x4
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(58) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(59) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U911(X1, X2, mark(X3), 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)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

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

The TRS R consists of the following rules:

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

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

(61) 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, x4)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
active1 > U91^12


The following usable rules [FROCOS05] were oriented: none

(62) 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(active(X1), X2, X3, X4) → U911(X1, X2, X3, X4)
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.

(63) 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(x2, x3, x4)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
active1 > U91^13


The following usable rules [FROCOS05] were oriented: none

(64) 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(active(X1), 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.

(65) 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)
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(x2, x3, x4)
mark(x1)  =  mark(x1)
active(x1)  =  active

Lexicographic Path Order [LPO].
Precedence:
mark1 > U91^13
active > U91^13


The following usable rules [FROCOS05] were oriented: none

(66) Obligation:

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

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

(67) 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)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U911(x1, x2, x3, x4)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(68) Obligation:

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

U911(active(X1), 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.

(69) 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)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(70) 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.

(71) PisEmptyProof (EQUIVALENT transformation)

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

(72) TRUE

(73) 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.

(74) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(75) Obligation:

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

U811(mark(X)) → U811(X)

The TRS R consists of the following rules:

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

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

(76) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(77) Obligation:

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

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

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

(78) PisEmptyProof (EQUIVALENT transformation)

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

(79) TRUE

(80) Obligation:

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

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.

(81) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(82) Obligation:

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

LENGTH(mark(X)) → LENGTH(X)

The TRS R consists of the following rules:

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


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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(84) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(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) PisEmptyProof (EQUIVALENT transformation)

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

(86) TRUE

(87) Obligation:

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

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.

(88) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(89) Obligation:

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

S(mark(X)) → S(X)

The TRS R consists of the following rules:

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


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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(91) Obligation:

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

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

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.

(95) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(96) Obligation:

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

ISNAT(mark(X)) → ISNAT(X)

The TRS R consists of the following rules:

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


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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(98) 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.

(99) PisEmptyProof (EQUIVALENT transformation)

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

(100) TRUE

(101) 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.

(102) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U721(X1, mark(X2)) → U721(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U721(x1, x2)  =  U721(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(103) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(104) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U721(X1, active(X2)) → U721(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U721(x1, x2)  =  x2
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(105) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(106) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U721(mark(X1), X2) → U721(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U721(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(107) Obligation:

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

U721(active(X1), X2) → U721(X1, X2)

The TRS R consists of the following rules:

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

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

(108) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U721(active(X1), X2) → U721(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U721(x1, x2)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(109) Obligation:

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

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

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

(110) PisEmptyProof (EQUIVALENT transformation)

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

(111) TRUE

(112) Obligation:

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

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.

(113) 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)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U711(x1, x2, x3)  =  U711(x3)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(114) Obligation:

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

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

(115) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U711(X1, X2, active(X3)) → U711(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U711(x1, x2, x3)  =  x3
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(116) 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(active(X1), 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.

(117) 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(x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(118) Obligation:

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

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

(119) 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)  =  U711(x2)
mark(x1)  =  mark
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(120) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(121) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U711(mark(X1), X2, X3) → U711(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U711(x1, x2, x3)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(122) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(123) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U711(active(X1), X2, X3) → U711(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U711(x1, x2, x3)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(124) 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.

(125) PisEmptyProof (EQUIVALENT transformation)

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

(126) TRUE

(127) 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.

(128) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(129) Obligation:

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

U621(mark(X)) → U621(X)

The TRS R consists of the following rules:

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


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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(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:

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.

(135) 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)  =  U611(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(136) 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.

(137) 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)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(138) 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.

(139) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(140) 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.

(141) 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)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(142) 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.

(143) PisEmptyProof (EQUIVALENT transformation)

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

(144) TRUE

(145) 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.

(146) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(147) Obligation:

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

ISNATLIST(mark(X)) → ISNATLIST(X)

The TRS R consists of the following rules:

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

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

(148) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(149) 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.

(150) PisEmptyProof (EQUIVALENT transformation)

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

(151) TRUE

(152) 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.

(153) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(154) Obligation:

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

U521(mark(X)) → U521(X)

The TRS R consists of the following rules:

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

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

(155) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(156) 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.

(157) PisEmptyProof (EQUIVALENT transformation)

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

(158) TRUE

(159) 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.

(160) 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)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U511(x1, x2)  =  U511(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(161) 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)
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.

(162) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(163) 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.

(164) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(165) 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.

(166) 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)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(167) 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.

(168) PisEmptyProof (EQUIVALENT transformation)

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

(169) TRUE

(170) 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.

(171) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(172) Obligation:

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

ISNATILIST(mark(X)) → ISNATILIST(X)

The TRS R consists of the following rules:

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

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

(173) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(174) 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.

(175) PisEmptyProof (EQUIVALENT transformation)

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

(176) TRUE

(177) 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.

(178) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(179) Obligation:

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

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.

(180) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(181) 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.

(182) PisEmptyProof (EQUIVALENT transformation)

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

(183) TRUE

(184) 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.

(185) 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)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U411(x1, x2)  =  U411(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(186) Obligation:

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

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.

(187) 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)  =  x2
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(188) Obligation:

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

U411(mark(X1), X2) → U411(X1, X2)
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.

(189) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(190) 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.

(191) 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)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(192) 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.

(193) PisEmptyProof (EQUIVALENT transformation)

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

(194) TRUE

(195) 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.

(196) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(197) Obligation:

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

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.

(198) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(199) 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.

(200) PisEmptyProof (EQUIVALENT transformation)

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

(201) TRUE

(202) 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.

(203) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(204) Obligation:

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

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.

(205) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(206) 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.

(207) PisEmptyProof (EQUIVALENT transformation)

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

(208) TRUE

(209) 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.

(210) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(211) Obligation:

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

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.

(212) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(213) 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.

(214) PisEmptyProof (EQUIVALENT transformation)

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

(215) TRUE

(216) 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.

(217) 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

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(218) Obligation:

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

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.

(219) 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
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(220) Obligation:

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

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

The TRS R consists of the following rules:

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

(221) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(222) 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.

(223) 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)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(224) 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.

(225) PisEmptyProof (EQUIVALENT transformation)

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

(226) TRUE

(227) 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.

(228) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  MARK
cons(x1, x2)  =  cons
ACTIVE(x1)  =  x1
mark(x1)  =  mark
U41(x1, x2)  =  U41
tt  =  tt
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
zeros  =  zeros
0  =  0
U11(x1)  =  U11
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U21(x1)  =  U21
U61(x1, x2)  =  U61
U62(x1)  =  U62
U31(x1)  =  U31
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take
U81(x1)  =  U81
active(x1)  =  active
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:
mark > 0 > tt > [MARK, U41, U42, isNatIList, zeros, U11, U51, U52, isNatList, U21, U61, U62, U31, U71, U72, isNat, s, length, U91, U92, U93, take, U81, active] > cons
mark > 0 > tt > [MARK, U41, U42, isNatIList, zeros, U11, U51, U52, isNatList, U21, U61, U62, U31, U71, U72, isNat, s, length, U91, U92, U93, take, U81, active] > nil


The following usable rules [FROCOS05] were oriented:

U42(active(X)) → U42(X)
U42(mark(X)) → U42(X)
U31(active(X)) → U31(X)
U31(mark(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U52(active(X)) → U52(X)
U52(mark(X)) → U52(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(X1, mark(X2)) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U21(active(X)) → U21(X)
U21(mark(X)) → U21(X)
U11(active(X)) → U11(X)
U11(mark(X)) → U11(X)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(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, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U81(active(X)) → U81(X)
U81(mark(X)) → U81(X)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)

(229) Obligation:

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

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.

(230) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
0 > tt > [U41, MARK, U42, isNatIList, zeros, U11, mark1, U51, U52, isNatList, U61, U31, U71, U72, isNat, length, U91, U92, U93, take] > active1 > U21 > s
0 > tt > [U41, MARK, U42, isNatIList, zeros, U11, mark1, U51, U52, isNatList, U61, U31, U71, U72, isNat, length, U91, U92, U93, take] > active1 > U62 > s
0 > tt > [U41, MARK, U42, isNatIList, zeros, U11, mark1, U51, U52, isNatList, U61, U31, U71, U72, isNat, length, U91, U92, U93, take] > active1 > [U81, nil] > s


The following usable rules [FROCOS05] were oriented:

U42(active(X)) → U42(X)
U42(mark(X)) → U42(X)
U31(active(X)) → U31(X)
U31(mark(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U52(active(X)) → U52(X)
U52(mark(X)) → U52(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(X1, mark(X2)) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U21(active(X)) → U21(X)
U21(mark(X)) → U21(X)
U11(active(X)) → U11(X)
U11(mark(X)) → U11(X)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(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, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U81(active(X)) → U81(X)
U81(mark(X)) → U81(X)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)

(231) Obligation:

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

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)
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)
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))
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)) → 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.

(232) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
0 > [U41, MARK, U42, isNatIList, zeros, U51, U52, isNatList, U61, U31, U71, U72, isNat, length, U91, U92, U93, take] > tt > [active1, nil] > mark > U11
0 > [U41, MARK, U42, isNatIList, zeros, U51, U52, isNatList, U61, U31, U71, U72, isNat, length, U91, U92, U93, take] > tt > [active1, nil] > mark > s
U621 > [U41, MARK, U42, isNatIList, zeros, U51, U52, isNatList, U61, U31, U71, U72, isNat, length, U91, U92, U93, take] > tt > [active1, nil] > mark > U11
U621 > [U41, MARK, U42, isNatIList, zeros, U51, U52, isNatList, U61, U31, U71, U72, isNat, length, U91, U92, U93, take] > tt > [active1, nil] > mark > s


The following usable rules [FROCOS05] were oriented:

U42(active(X)) → U42(X)
U42(mark(X)) → U42(X)
U31(active(X)) → U31(X)
U31(mark(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U52(active(X)) → U52(X)
U52(mark(X)) → U52(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(X1, mark(X2)) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U11(active(X)) → U11(X)
U11(mark(X)) → U11(X)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(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, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
length(active(X)) → length(X)
length(mark(X)) → length(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)

(233) Obligation:

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

ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(zeros) → ACTIVE(zeros)
ACTIVE(zeros) → MARK(cons(0, zeros))
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
MARK(U11(X)) → 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)
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))
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)) → 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.

(234) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
[U81, active] > cons > [tt, mark] > [U41, MARK, isNatIList, zeros, U51, U52, isNatList, U61, U31, U71, U72, isNat, length, U91, U92, U93, take] > U42 > U11
[U81, active] > cons > [tt, mark] > [U41, MARK, isNatIList, zeros, U51, U52, isNatList, U61, U31, U71, U72, isNat, length, U91, U92, U93, take] > s > U11
[U81, active] > U21 > [tt, mark] > [U41, MARK, isNatIList, zeros, U51, U52, isNatList, U61, U31, U71, U72, isNat, length, U91, U92, U93, take] > U42 > U11
[U81, active] > U21 > [tt, mark] > [U41, MARK, isNatIList, zeros, U51, U52, isNatList, U61, U31, U71, U72, isNat, length, U91, U92, U93, take] > s > U11
[U81, active] > nil > 0 > [tt, mark] > [U41, MARK, isNatIList, zeros, U51, U52, isNatList, U61, U31, U71, U72, isNat, length, U91, U92, U93, take] > U42 > U11
[U81, active] > nil > 0 > [tt, mark] > [U41, MARK, isNatIList, zeros, U51, U52, isNatList, U61, U31, U71, U72, isNat, length, U91, U92, U93, take] > s > U11


The following usable rules [FROCOS05] were oriented:

U42(active(X)) → U42(X)
U42(mark(X)) → U42(X)
U31(active(X)) → U31(X)
U31(mark(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U52(active(X)) → U52(X)
U52(mark(X)) → U52(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(X1, mark(X2)) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(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, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
length(active(X)) → length(X)
length(mark(X)) → length(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)

(235) Obligation:

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

ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(zeros) → ACTIVE(zeros)
ACTIVE(zeros) → MARK(cons(0, zeros))
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
MARK(U11(X)) → 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)
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)
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))
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)) → 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.

(236) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
U21 > [U41, MARK, isNatIList, zeros, U51, U52, isNatList, U61, U71, U72, isNat, length, U91, U92, U93, take] > tt > cons2 > mark > s > active
U21 > [U41, MARK, isNatIList, zeros, U51, U52, isNatList, U61, U71, U72, isNat, length, U91, U92, U93, take] > tt > U62 > mark > s > active
U21 > [U41, MARK, isNatIList, zeros, U51, U52, isNatList, U61, U71, U72, isNat, length, U91, U92, U93, take] > 0 > mark > s > active
U21 > [U41, MARK, isNatIList, zeros, U51, U52, isNatList, U61, U71, U72, isNat, length, U91, U92, U93, take] > U31 > mark > s > active
U811 > [U41, MARK, isNatIList, zeros, U51, U52, isNatList, U61, U71, U72, isNat, length, U91, U92, U93, take] > tt > cons2 > mark > s > active
U811 > [U41, MARK, isNatIList, zeros, U51, U52, isNatList, U61, U71, U72, isNat, length, U91, U92, U93, take] > tt > U62 > mark > s > active
U811 > [U41, MARK, isNatIList, zeros, U51, U52, isNatList, U61, U71, U72, isNat, length, U91, U92, U93, take] > 0 > mark > s > active
U811 > [U41, MARK, isNatIList, zeros, U51, U52, isNatList, U61, U71, U72, isNat, length, U91, U92, U93, take] > U31 > mark > s > active
U811 > nil > tt > cons2 > mark > s > active
U811 > nil > tt > U62 > mark > s > active
U811 > nil > 0 > mark > s > active


The following usable rules [FROCOS05] were oriented:

U31(active(X)) → U31(X)
U31(mark(X)) → U31(X)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U52(active(X)) → U52(X)
U52(mark(X)) → U52(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(X1, mark(X2)) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(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, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)

(237) Obligation:

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

ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(zeros) → ACTIVE(zeros)
ACTIVE(zeros) → MARK(cons(0, zeros))
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
MARK(U11(X)) → MARK(X)
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
MARK(U21(X)) → 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)
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)
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))
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)) → 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.

(238) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
U421 > [tt, U11] > nil > 0 > [U41, MARK, isNatIList, zeros, U51, isNatList, U61, U71, U72, isNat, length, U91, U92, U93, take] > [s, active1] > mark > U52
U421 > [tt, U11] > nil > 0 > [U41, MARK, isNatIList, zeros, U51, isNatList, U61, U71, U72, isNat, length, U91, U92, U93, take] > [s, active1] > mark > U211
U31 > [tt, U11] > nil > 0 > [U41, MARK, isNatIList, zeros, U51, isNatList, U61, U71, U72, isNat, length, U91, U92, U93, take] > [s, active1] > mark > U52
U31 > [tt, U11] > nil > 0 > [U41, MARK, isNatIList, zeros, U51, isNatList, U61, U71, U72, isNat, length, U91, U92, U93, take] > [s, active1] > mark > U211
U811 > [U41, MARK, isNatIList, zeros, U51, isNatList, U61, U71, U72, isNat, length, U91, U92, U93, take] > [s, active1] > mark > U52
U811 > [U41, MARK, isNatIList, zeros, U51, isNatList, U61, U71, U72, isNat, length, U91, U92, U93, take] > [s, active1] > mark > U211


The following usable rules [FROCOS05] were oriented:

U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U52(active(X)) → U52(X)
U52(mark(X)) → U52(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(X1, mark(X2)) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(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, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)

(239) Obligation:

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

ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(zeros) → ACTIVE(zeros)
ACTIVE(zeros) → MARK(cons(0, zeros))
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
MARK(U11(X)) → MARK(X)
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
MARK(U21(X)) → 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)
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)
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)
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))
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)) → 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.