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

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 AND
  ↳5 QDP
    ↳6 QDPOrderProof (⇔)
    ↳7 QDP
    ↳8 PisEmptyProof (⇔)
    ↳9 TRUE
  ↳10 QDP
    ↳11 QDPOrderProof (⇔)
    ↳12 QDP
    ↳13 PisEmptyProof (⇔)
    ↳14 TRUE
  ↳15 QDP
    ↳16 QDPOrderProof (⇔)
    ↳17 QDP
    ↳18 PisEmptyProof (⇔)
    ↳19 TRUE
  ↳20 QDP
    ↳21 QDPOrderProof (⇔)
    ↳22 QDP
    ↳23 QDPOrderProof (⇔)
    ↳24 QDP
    ↳25 QDPOrderProof (⇔)
    ↳26 QDP
    ↳27 PisEmptyProof (⇔)
    ↳28 TRUE
  ↳29 QDP
    ↳30 QDPOrderProof (⇔)
    ↳31 QDP
    ↳32 QDPOrderProof (⇔)
    ↳33 QDP
    ↳34 PisEmptyProof (⇔)
    ↳35 TRUE
  ↳36 QDP
    ↳37 QDPOrderProof (⇔)
    ↳38 QDP
    ↳39 QDPOrderProof (⇔)
    ↳40 QDP
    ↳41 PisEmptyProof (⇔)
    ↳42 TRUE
  ↳43 QDP
    ↳44 QDPOrderProof (⇔)
    ↳45 QDP
    ↳46 QDPOrderProof (⇔)
    ↳47 QDP
    ↳48 PisEmptyProof (⇔)
    ↳49 TRUE
  ↳50 QDP
    ↳51 QDPOrderProof (⇔)
    ↳52 QDP
    ↳53 QDPOrderProof (⇔)
    ↳54 QDP
    ↳55 PisEmptyProof (⇔)
    ↳56 TRUE
  ↳57 QDP
    ↳58 QDPOrderProof (⇔)
    ↳59 QDP
    ↳60 QDPOrderProof (⇔)
    ↳61 QDP
    ↳62 PisEmptyProof (⇔)
    ↳63 TRUE
  ↳64 QDP
    ↳65 QDPOrderProof (⇔)
    ↳66 QDP
    ↳67 QDPOrderProof (⇔)
    ↳68 QDP
    ↳69 PisEmptyProof (⇔)
    ↳70 TRUE
  ↳71 QDP
    ↳72 QDPOrderProof (⇔)
    ↳73 QDP
    ↳74 QDPOrderProof (⇔)
    ↳75 QDP
    ↳76 PisEmptyProof (⇔)
    ↳77 TRUE
  ↳78 QDP
    ↳79 QDPOrderProof (⇔)
    ↳80 QDP
    ↳81 QDPOrderProof (⇔)
    ↳82 QDP
    ↳83 PisEmptyProof (⇔)
    ↳84 TRUE
  ↳85 QDP
    ↳86 QDPOrderProof (⇔)
    ↳87 QDP
    ↳88 QDPOrderProof (⇔)
    ↳89 QDP
    ↳90 PisEmptyProof (⇔)
    ↳91 TRUE
  ↳92 QDP
    ↳93 QDPOrderProof (⇔)
    ↳94 QDP
    ↳95 QDPOrderProof (⇔)
    ↳96 QDP
    ↳97 PisEmptyProof (⇔)
    ↳98 TRUE
  ↳99 QDP
    ↳100 QDPOrderProof (⇔)
    ↳101 QDP
    ↳102 QDPOrderProof (⇔)
    ↳103 QDP
    ↳104 PisEmptyProof (⇔)
    ↳105 TRUE
  ↳106 QDP
    ↳107 QDPOrderProof (⇔)
    ↳108 QDP
    ↳109 QDPOrderProof (⇔)
    ↳110 QDP
    ↳111 PisEmptyProof (⇔)
    ↳112 TRUE
  ↳113 QDP
    ↳114 QDPOrderProof (⇔)
    ↳115 QDP
    ↳116 QDPOrderProof (⇔)
    ↳117 QDP
    ↳118 PisEmptyProof (⇔)
    ↳119 TRUE
  ↳120 QDP
    ↳121 QDPOrderProof (⇔)
    ↳122 QDP
    ↳123 QDPOrderProof (⇔)
    ↳124 QDP
    ↳125 PisEmptyProof (⇔)
    ↳126 TRUE
  ↳127 QDP
    ↳128 QDPOrderProof (⇔)
    ↳129 QDP
    ↳130 QDPOrderProof (⇔)
    ↳131 QDP
    ↳132 PisEmptyProof (⇔)
    ↳133 TRUE
  ↳134 QDP
    ↳135 QDPOrderProof (⇔)
    ↳136 QDP
    ↳137 QDPOrderProof (⇔)
    ↳138 QDP
    ↳139 PisEmptyProof (⇔)
    ↳140 TRUE
  ↳141 QDP
    ↳142 QDPOrderProof (⇔)
    ↳143 QDP
    ↳144 QDPOrderProof (⇔)
    ↳145 QDP
    ↳146 PisEmptyProof (⇔)
    ↳147 TRUE
  ↳148 QDP
    ↳149 QDPOrderProof (⇔)
    ↳150 QDP
    ↳151 QDPOrderProof (⇔)
    ↳152 QDP
    ↳153 PisEmptyProof (⇔)
    ↳154 TRUE
  ↳155 QDP
    ↳156 QDPOrderProof (⇔)
    ↳157 QDP
    ↳158 QDPOrderProof (⇔)
    ↳159 QDP
    ↳160 QDPOrderProof (⇔)
    ↳161 QDP
    ↳162 QDPOrderProof (⇔)
    ↳163 QDP
    ↳164 QDPOrderProof (⇔)
    ↳165 QDP
    ↳166 QDPOrderProof (⇔)
    ↳167 QDP
    ↳168 QDPOrderProof (⇔)
    ↳169 QDP
    ↳170 QDPOrderProof (⇔)
    ↳171 QDP
    ↳172 QDPOrderProof (⇔)
    ↳173 QDP
    ↳174 QDPOrderProof (⇔)
    ↳175 QDP
    ↳176 PisEmptyProof (⇔)
    ↳177 TRUE
  ↳178 QDP
    ↳179 QDPOrderProof (⇔)
    ↳180 QDP
    ↳181 QDPOrderProof (⇔)
    ↳182 QDP
    ↳183 QDPOrderProof (⇔)
    ↳184 QDP
    ↳185 PisEmptyProof (⇔)
    ↳186 TRUE
  ↳187 QDP

(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

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

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

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 25 SCCs with 79 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

ISNAT(ok(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ISNAT(ok(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
ok(x1)  =  ok(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
ok1: [1]


The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

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

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

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

(8) PisEmptyProof (EQUIVALENT transformation)

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

(9) TRUE

(10) Obligation:

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

ISNATLIST(ok(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ISNATLIST(ok(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
ok(x1)  =  ok(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
ok1: [1]


The following usable rules [FROCOS05] were oriented: none

(12) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(13) PisEmptyProof (EQUIVALENT transformation)

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

(14) TRUE

(15) Obligation:

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

ISNATILIST(ok(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(16) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ISNATILIST(ok(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
ok(x1)  =  ok(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
ok1: [1]


The following usable rules [FROCOS05] were oriented: none

(17) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(18) PisEmptyProof (EQUIVALENT transformation)

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

(19) TRUE

(20) 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(ok(X1), ok(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.


TAKE(ok(X1), ok(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)  =  x1
ok(x1)  =  ok(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
[TAKE1, ok1]

Status:
TAKE1: [1]
ok1: [1]


The following usable rules [FROCOS05] were oriented: none

(22) 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)

The TRS R consists of the following rules:

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

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.


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)

Lexicographic path order with status [LPO].
Quasi-Precedence:
mark1 > TAKE1

Status:
TAKE1: [1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(24) Obligation:

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

TAKE(mark(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.


TAKE(mark(X1), X2) → TAKE(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:
mark1 > TAKE2

Status:
TAKE2: [2,1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(26) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(27) PisEmptyProof (EQUIVALENT transformation)

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

(28) TRUE

(29) Obligation:

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

U931(ok(X1), ok(X2), ok(X3), ok(X4)) → U931(X1, X2, X3, X4)
U931(mark(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(30) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U931(ok(X1), ok(X2), ok(X3), ok(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)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:
[U93^11, ok1]

Status:
U93^11: [1]
ok1: [1]


The following usable rules [FROCOS05] were oriented: none

(31) Obligation:

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

U931(mark(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(32) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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)

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(33) Obligation:

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

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

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

(34) PisEmptyProof (EQUIVALENT transformation)

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

(35) TRUE

(36) Obligation:

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

U921(ok(X1), ok(X2), ok(X3), ok(X4)) → U921(X1, X2, X3, X4)
U921(mark(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(37) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U921(ok(X1), ok(X2), ok(X3), ok(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)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:
[U92^11, ok1]

Status:
ok1: [1]
U92^11: [1]


The following usable rules [FROCOS05] were oriented: none

(38) Obligation:

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

U921(mark(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(39) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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)

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(40) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(41) PisEmptyProof (EQUIVALENT transformation)

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

(42) TRUE

(43) Obligation:

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

U911(ok(X1), ok(X2), ok(X3), ok(X4)) → U911(X1, X2, X3, X4)
U911(mark(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(44) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic path order with status [LPO].
Quasi-Precedence:
[U91^11, ok1]

Status:
U91^11: [1]
ok1: [1]


The following usable rules [FROCOS05] were oriented: none

(45) Obligation:

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

U911(mark(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.


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)

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(47) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(48) PisEmptyProof (EQUIVALENT transformation)

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

(49) TRUE

(50) Obligation:

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

U811(ok(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(51) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic path order with status [LPO].
Quasi-Precedence:
[U81^11, ok1]

Status:
U81^11: [1]
ok1: [1]


The following usable rules [FROCOS05] were oriented: none

(52) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(53) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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 with status [LPO].
Quasi-Precedence:
trivial

Status:
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(54) Obligation:

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

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

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

(55) PisEmptyProof (EQUIVALENT transformation)

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

(56) TRUE

(57) Obligation:

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

LENGTH(ok(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(58) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic path order with status [LPO].
Quasi-Precedence:
[LENGTH1, ok1]

Status:
LENGTH1: [1]
ok1: [1]


The following usable rules [FROCOS05] were oriented: none

(59) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(60) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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 with status [LPO].
Quasi-Precedence:
trivial

Status:
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(61) Obligation:

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

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

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

(62) PisEmptyProof (EQUIVALENT transformation)

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

(63) TRUE

(64) Obligation:

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

S(ok(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(65) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic path order with status [LPO].
Quasi-Precedence:
[S1, ok1]

Status:
ok1: [1]
S1: [1]


The following usable rules [FROCOS05] were oriented: none

(66) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(67) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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 with status [LPO].
Quasi-Precedence:
trivial

Status:
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(68) Obligation:

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

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

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

(69) PisEmptyProof (EQUIVALENT transformation)

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

(70) TRUE

(71) Obligation:

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

U721(ok(X1), ok(X2)) → U721(X1, X2)
U721(mark(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(72) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U721(ok(X1), ok(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)
ok(x1)  =  ok(x1)
mark(x1)  =  mark

Lexicographic path order with status [LPO].
Quasi-Precedence:
ok1 > [U72^11, mark]

Status:
mark: []
ok1: [1]
U72^11: [1]


The following usable rules [FROCOS05] were oriented: none

(73) Obligation:

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

U721(mark(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(74) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U721(mark(X1), X2) → U721(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:
mark1 > U72^12

Status:
mark1: [1]
U72^12: [2,1]


The following usable rules [FROCOS05] were oriented: none

(75) Obligation:

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

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

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

(76) PisEmptyProof (EQUIVALENT transformation)

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

(77) TRUE

(78) Obligation:

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

U711(ok(X1), ok(X2), ok(X3)) → U711(X1, X2, X3)
U711(mark(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(79) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U711(ok(X1), ok(X2), ok(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)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:
ok1 > U71^11

Status:
U71^11: [1]
ok1: [1]


The following usable rules [FROCOS05] were oriented: none

(80) Obligation:

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

U711(mark(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(81) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U711(mark(X1), X2, X3) → U711(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:
[U71^13, mark1]

Status:
U71^13: [3,2,1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(82) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(83) PisEmptyProof (EQUIVALENT transformation)

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

(84) TRUE

(85) Obligation:

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

U621(ok(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(86) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic path order with status [LPO].
Quasi-Precedence:
[U62^11, ok1]

Status:
ok1: [1]
U62^11: [1]


The following usable rules [FROCOS05] were oriented: none

(87) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(88) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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 with status [LPO].
Quasi-Precedence:
trivial

Status:
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(89) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(90) PisEmptyProof (EQUIVALENT transformation)

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

(91) TRUE

(92) Obligation:

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

U611(ok(X1), ok(X2)) → U611(X1, X2)
U611(mark(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(93) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U611(ok(X1), ok(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)
ok(x1)  =  ok(x1)
mark(x1)  =  mark

Lexicographic path order with status [LPO].
Quasi-Precedence:
ok1 > [U61^11, mark]

Status:
U61^11: [1]
mark: []
ok1: [1]


The following usable rules [FROCOS05] were oriented: none

(94) Obligation:

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

U611(mark(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(95) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U611(mark(X1), X2) → U611(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:
mark1 > U61^12

Status:
U61^12: [2,1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(96) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(97) PisEmptyProof (EQUIVALENT transformation)

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

(98) TRUE

(99) Obligation:

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

U521(ok(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(100) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic path order with status [LPO].
Quasi-Precedence:
[U52^11, ok1]

Status:
U52^11: [1]
ok1: [1]


The following usable rules [FROCOS05] were oriented: none

(101) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.


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 with status [LPO].
Quasi-Precedence:
trivial

Status:
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(103) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(104) PisEmptyProof (EQUIVALENT transformation)

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

(105) TRUE

(106) Obligation:

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

U511(ok(X1), ok(X2)) → U511(X1, X2)
U511(mark(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(107) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U511(ok(X1), ok(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)
ok(x1)  =  ok(x1)
mark(x1)  =  mark

Lexicographic path order with status [LPO].
Quasi-Precedence:
ok1 > [U51^11, mark]

Status:
mark: []
ok1: [1]
U51^11: [1]


The following usable rules [FROCOS05] were oriented: none

(108) Obligation:

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

U511(mark(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(109) 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: Lexicographic path order with status [LPO].
Quasi-Precedence:
mark1 > U51^12

Status:
U51^12: [2,1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(110) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(111) PisEmptyProof (EQUIVALENT transformation)

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

(112) TRUE

(113) Obligation:

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

U421(ok(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(114) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic path order with status [LPO].
Quasi-Precedence:
[U42^11, ok1]

Status:
ok1: [1]
U42^11: [1]


The following usable rules [FROCOS05] were oriented: none

(115) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(116) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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 with status [LPO].
Quasi-Precedence:
trivial

Status:
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(117) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(118) PisEmptyProof (EQUIVALENT transformation)

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

(119) TRUE

(120) Obligation:

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

U411(ok(X1), ok(X2)) → U411(X1, X2)
U411(mark(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.


U411(ok(X1), ok(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)
ok(x1)  =  ok(x1)
mark(x1)  =  mark

Lexicographic path order with status [LPO].
Quasi-Precedence:
ok1 > [U41^11, mark]

Status:
U41^11: [1]
mark: []
ok1: [1]


The following usable rules [FROCOS05] were oriented: none

(122) Obligation:

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

U411(mark(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.


U411(mark(X1), X2) → U411(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:
mark1 > U41^12

Status:
U41^12: [2,1]
mark1: [1]


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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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:

U311(ok(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.


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

Lexicographic path order with status [LPO].
Quasi-Precedence:
[U31^11, ok1]

Status:
U31^11: [1]
ok1: [1]


The following usable rules [FROCOS05] were oriented: none

(129) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.


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 with status [LPO].
Quasi-Precedence:
trivial

Status:
mark1: [1]


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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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:

U211(ok(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.


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

Lexicographic path order with status [LPO].
Quasi-Precedence:
[U21^11, ok1]

Status:
ok1: [1]
U21^11: [1]


The following usable rules [FROCOS05] were oriented: none

(136) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.


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 with status [LPO].
Quasi-Precedence:
trivial

Status:
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(138) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(139) PisEmptyProof (EQUIVALENT transformation)

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

(140) TRUE

(141) Obligation:

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

U111(ok(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(142) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic path order with status [LPO].
Quasi-Precedence:
[U11^11, ok1]

Status:
U11^11: [1]
ok1: [1]


The following usable rules [FROCOS05] were oriented: none

(143) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(144) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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 with status [LPO].
Quasi-Precedence:
trivial

Status:
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(145) Obligation:

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

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

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

(146) PisEmptyProof (EQUIVALENT transformation)

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

(147) TRUE

(148) Obligation:

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

CONS(ok(X1), ok(X2)) → CONS(X1, X2)
CONS(mark(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(149) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


CONS(ok(X1), ok(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)
ok(x1)  =  ok(x1)
mark(x1)  =  mark

Lexicographic path order with status [LPO].
Quasi-Precedence:
ok1 > [CONS1, mark]

Status:
CONS1: [1]
mark: []
ok1: [1]


The following usable rules [FROCOS05] were oriented: none

(150) Obligation:

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

CONS(mark(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(151) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


CONS(mark(X1), X2) → CONS(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:
mark1 > CONS2

Status:
CONS2: [2,1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(152) Obligation:

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

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

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

(153) PisEmptyProof (EQUIVALENT transformation)

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

(154) TRUE

(155) Obligation:

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

PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(U11(X)) → PROPER(X)
PROPER(U21(X)) → PROPER(X)
PROPER(U31(X)) → PROPER(X)
PROPER(U41(X1, X2)) → PROPER(X1)
PROPER(U41(X1, X2)) → PROPER(X2)
PROPER(U42(X)) → PROPER(X)
PROPER(isNatIList(X)) → PROPER(X)
PROPER(U51(X1, X2)) → PROPER(X1)
PROPER(U51(X1, X2)) → PROPER(X2)
PROPER(U52(X)) → PROPER(X)
PROPER(isNatList(X)) → PROPER(X)
PROPER(U61(X1, X2)) → PROPER(X1)
PROPER(U61(X1, X2)) → PROPER(X2)
PROPER(U62(X)) → PROPER(X)
PROPER(U71(X1, X2, X3)) → PROPER(X1)
PROPER(U71(X1, X2, X3)) → PROPER(X2)
PROPER(U71(X1, X2, X3)) → PROPER(X3)
PROPER(U72(X1, X2)) → PROPER(X1)
PROPER(U72(X1, X2)) → PROPER(X2)
PROPER(isNat(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → PROPER(X)
PROPER(U81(X)) → PROPER(X)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X1)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X2)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X3)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X4)
PROPER(U92(X1, X2, X3, X4)) → PROPER(X1)
PROPER(U92(X1, X2, X3, X4)) → PROPER(X2)
PROPER(U92(X1, X2, X3, X4)) → PROPER(X3)
PROPER(U92(X1, X2, X3, X4)) → PROPER(X4)
PROPER(U93(X1, X2, X3, X4)) → PROPER(X1)
PROPER(U93(X1, X2, X3, X4)) → PROPER(X2)
PROPER(U93(X1, X2, X3, X4)) → PROPER(X3)
PROPER(U93(X1, X2, X3, X4)) → PROPER(X4)
PROPER(take(X1, X2)) → PROPER(X1)
PROPER(take(X1, X2)) → PROPER(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(156) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(U41(X1, X2)) → PROPER(X1)
PROPER(U41(X1, X2)) → PROPER(X2)
PROPER(U51(X1, X2)) → PROPER(X1)
PROPER(U51(X1, X2)) → PROPER(X2)
PROPER(U61(X1, X2)) → PROPER(X1)
PROPER(U61(X1, X2)) → PROPER(X2)
PROPER(U71(X1, X2, X3)) → PROPER(X1)
PROPER(U71(X1, X2, X3)) → PROPER(X2)
PROPER(U71(X1, X2, X3)) → PROPER(X3)
PROPER(U72(X1, X2)) → PROPER(X1)
PROPER(U72(X1, X2)) → PROPER(X2)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X1)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X2)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X3)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X4)
PROPER(U92(X1, X2, X3, X4)) → PROPER(X1)
PROPER(U92(X1, X2, X3, X4)) → PROPER(X2)
PROPER(U92(X1, X2, X3, X4)) → PROPER(X3)
PROPER(U92(X1, X2, X3, X4)) → PROPER(X4)
PROPER(U93(X1, X2, X3, X4)) → PROPER(X1)
PROPER(U93(X1, X2, X3, X4)) → PROPER(X2)
PROPER(U93(X1, X2, X3, X4)) → PROPER(X3)
PROPER(U93(X1, X2, X3, X4)) → PROPER(X4)
PROPER(take(X1, X2)) → PROPER(X1)
PROPER(take(X1, X2)) → PROPER(X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
cons(x1, x2)  =  cons(x1, x2)
U11(x1)  =  x1
U21(x1)  =  x1
U31(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  x1
isNatIList(x1)  =  x1
U51(x1, x2)  =  U51(x1, x2)
U52(x1)  =  x1
isNatList(x1)  =  x1
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  x1
U71(x1, x2, x3)  =  U71(x1, x2, x3)
U72(x1, x2)  =  U72(x1, x2)
isNat(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1
U81(x1)  =  x1
U91(x1, x2, x3, x4)  =  U91(x1, x2, x3, x4)
U92(x1, x2, x3, x4)  =  U92(x1, x2, x3, x4)
U93(x1, x2, x3, x4)  =  U93(x1, x2, x3, x4)
take(x1, x2)  =  take(x1, x2)

Lexicographic path order with status [LPO].
Quasi-Precedence:
U722 > PROPER1
U934 > PROPER1

Status:
PROPER1: [1]
cons2: [2,1]
U914: [4,1,2,3]
U612: [1,2]
U412: [1,2]
U934: [2,3,1,4]
U722: [1,2]
U924: [4,3,1,2]
take2: [1,2]
U512: [2,1]
U713: [2,1,3]


The following usable rules [FROCOS05] were oriented: none

(157) Obligation:

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

PROPER(U11(X)) → PROPER(X)
PROPER(U21(X)) → PROPER(X)
PROPER(U31(X)) → PROPER(X)
PROPER(U42(X)) → PROPER(X)
PROPER(isNatIList(X)) → PROPER(X)
PROPER(U52(X)) → PROPER(X)
PROPER(isNatList(X)) → PROPER(X)
PROPER(U62(X)) → PROPER(X)
PROPER(isNat(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → PROPER(X)
PROPER(U81(X)) → PROPER(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(158) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(U11(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
U11(x1)  =  U11(x1)
U21(x1)  =  x1
U31(x1)  =  x1
U42(x1)  =  x1
isNatIList(x1)  =  x1
U52(x1)  =  x1
isNatList(x1)  =  x1
U62(x1)  =  x1
isNat(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1
U81(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
PROPER1: [1]
U111: [1]


The following usable rules [FROCOS05] were oriented: none

(159) Obligation:

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

PROPER(U21(X)) → PROPER(X)
PROPER(U31(X)) → PROPER(X)
PROPER(U42(X)) → PROPER(X)
PROPER(isNatIList(X)) → PROPER(X)
PROPER(U52(X)) → PROPER(X)
PROPER(isNatList(X)) → PROPER(X)
PROPER(U62(X)) → PROPER(X)
PROPER(isNat(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → PROPER(X)
PROPER(U81(X)) → PROPER(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.


PROPER(isNatList(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
U21(x1)  =  x1
U31(x1)  =  x1
U42(x1)  =  x1
isNatIList(x1)  =  x1
U52(x1)  =  x1
isNatList(x1)  =  isNatList(x1)
U62(x1)  =  x1
isNat(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1
U81(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:
isNatList1 > PROPER1

Status:
PROPER1: [1]
isNatList1: [1]


The following usable rules [FROCOS05] were oriented: none

(161) Obligation:

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

PROPER(U21(X)) → PROPER(X)
PROPER(U31(X)) → PROPER(X)
PROPER(U42(X)) → PROPER(X)
PROPER(isNatIList(X)) → PROPER(X)
PROPER(U52(X)) → PROPER(X)
PROPER(U62(X)) → PROPER(X)
PROPER(isNat(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → PROPER(X)
PROPER(U81(X)) → PROPER(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.


PROPER(U21(X)) → PROPER(X)
PROPER(U81(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
U21(x1)  =  U21(x1)
U31(x1)  =  x1
U42(x1)  =  x1
isNatIList(x1)  =  x1
U52(x1)  =  x1
U62(x1)  =  x1
isNat(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1
U81(x1)  =  U81(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
U811 > PROPER1

Status:
PROPER1: [1]
U811: [1]
U211: [1]


The following usable rules [FROCOS05] were oriented: none

(163) Obligation:

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

PROPER(U31(X)) → PROPER(X)
PROPER(U42(X)) → PROPER(X)
PROPER(isNatIList(X)) → PROPER(X)
PROPER(U52(X)) → PROPER(X)
PROPER(U62(X)) → PROPER(X)
PROPER(isNat(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → PROPER(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.


PROPER(U31(X)) → PROPER(X)
PROPER(U42(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
U31(x1)  =  U31(x1)
U42(x1)  =  U42(x1)
isNatIList(x1)  =  x1
U52(x1)  =  x1
U62(x1)  =  x1
isNat(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:
[PROPER1, U311]

Status:
PROPER1: [1]
U311: [1]
U421: [1]


The following usable rules [FROCOS05] were oriented: none

(165) Obligation:

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

PROPER(isNatIList(X)) → PROPER(X)
PROPER(U52(X)) → PROPER(X)
PROPER(U62(X)) → PROPER(X)
PROPER(isNat(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → PROPER(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.


PROPER(isNatIList(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
isNatIList(x1)  =  isNatIList(x1)
U52(x1)  =  x1
U62(x1)  =  x1
isNat(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
PROPER1: [1]
isNatIList1: [1]


The following usable rules [FROCOS05] were oriented: none

(167) Obligation:

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

PROPER(U52(X)) → PROPER(X)
PROPER(U62(X)) → PROPER(X)
PROPER(isNat(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → PROPER(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(168) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(U52(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
U52(x1)  =  U52(x1)
U62(x1)  =  x1
isNat(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
PROPER1: [1]
U521: [1]


The following usable rules [FROCOS05] were oriented: none

(169) Obligation:

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

PROPER(U62(X)) → PROPER(X)
PROPER(isNat(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → PROPER(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(170) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(isNat(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
U62(x1)  =  x1
isNat(x1)  =  isNat(x1)
s(x1)  =  s(x1)
length(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
PROPER1: [1]
s1: [1]
isNat1: [1]


The following usable rules [FROCOS05] were oriented: none

(171) Obligation:

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

PROPER(U62(X)) → PROPER(X)
PROPER(length(X)) → PROPER(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(172) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(U62(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
U62(x1)  =  U62(x1)
length(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:
[PROPER1, U621]

Status:
U621: [1]
PROPER1: [1]


The following usable rules [FROCOS05] were oriented: none

(173) Obligation:

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

PROPER(length(X)) → PROPER(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(174) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
length1: [1]


The following usable rules [FROCOS05] were oriented: none

(175) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(176) PisEmptyProof (EQUIVALENT transformation)

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

(177) TRUE

(178) Obligation:

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

ACTIVE(U11(X)) → ACTIVE(X)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(U21(X)) → ACTIVE(X)
ACTIVE(U31(X)) → ACTIVE(X)
ACTIVE(U41(X1, X2)) → ACTIVE(X1)
ACTIVE(U42(X)) → ACTIVE(X)
ACTIVE(U51(X1, X2)) → ACTIVE(X1)
ACTIVE(U52(X)) → ACTIVE(X)
ACTIVE(U61(X1, X2)) → ACTIVE(X1)
ACTIVE(U62(X)) → ACTIVE(X)
ACTIVE(U71(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U72(X1, X2)) → ACTIVE(X1)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(length(X)) → ACTIVE(X)
ACTIVE(U81(X)) → ACTIVE(X)
ACTIVE(U91(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(U92(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(U93(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(179) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U11(X)) → ACTIVE(X)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(U21(X)) → ACTIVE(X)
ACTIVE(U31(X)) → ACTIVE(X)
ACTIVE(U41(X1, X2)) → ACTIVE(X1)
ACTIVE(U42(X)) → ACTIVE(X)
ACTIVE(U51(X1, X2)) → ACTIVE(X1)
ACTIVE(U52(X)) → ACTIVE(X)
ACTIVE(U61(X1, X2)) → ACTIVE(X1)
ACTIVE(U62(X)) → ACTIVE(X)
ACTIVE(U71(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U72(X1, X2)) → ACTIVE(X1)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(U91(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(U92(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(U93(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  x1
U11(x1)  =  U11(x1)
cons(x1, x2)  =  cons(x1, x2)
U21(x1)  =  U21(x1)
U31(x1)  =  U31(x1)
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  U42(x1)
U51(x1, x2)  =  U51(x1)
U52(x1)  =  U52(x1)
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  U62(x1)
U71(x1, x2, x3)  =  U71(x1, x2, x3)
U72(x1, x2)  =  U72(x1, x2)
s(x1)  =  s(x1)
length(x1)  =  x1
U81(x1)  =  x1
U91(x1, x2, x3, x4)  =  U91(x1, x2, x4)
U92(x1, x2, x3, x4)  =  U92(x1, x2, x3)
U93(x1, x2, x3, x4)  =  U93(x1, x2, x3)
take(x1, x2)  =  take(x1, x2)

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
U722: [2,1]
take2: [1,2]
U621: [1]
cons2: [2,1]
U913: [3,1,2]
U311: [1]
U612: [2,1]
U933: [3,2,1]
U412: [2,1]
U421: [1]
U521: [1]
U923: [2,3,1]
U111: [1]
s1: [1]
U211: [1]
U511: [1]
U713: [2,3,1]


The following usable rules [FROCOS05] were oriented: none

(180) Obligation:

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

ACTIVE(length(X)) → ACTIVE(X)
ACTIVE(U81(X)) → ACTIVE(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(181) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(length(X)) → ACTIVE(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
length(x1)  =  length(x1)
U81(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:
[ACTIVE1, length1]

Status:
length1: [1]
ACTIVE1: [1]


The following usable rules [FROCOS05] were oriented: none

(182) Obligation:

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

ACTIVE(U81(X)) → ACTIVE(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(183) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U81(X)) → ACTIVE(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  x1
U81(x1)  =  U81(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
U811: [1]


The following usable rules [FROCOS05] were oriented: none

(184) Obligation:

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

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

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

(185) PisEmptyProof (EQUIVALENT transformation)

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

(186) TRUE

(187) Obligation:

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

TOP(ok(X)) → TOP(active(X))
TOP(mark(X)) → TOP(proper(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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