(STRATEGY
    INNERMOST)

(VAR
    IL L M N V V1 V2 X X1 X2 X3 X4)
(DATATYPES
    A = µX.< zeros, mark(X), 0, tt, nil, ok(X) >)
(SIGNATURES
    active :: [A] -> A
    cons :: [A x A] -> A
    U11 :: [A] -> A
    U21 :: [A] -> A
    U31 :: [A] -> A
    U41 :: [A x A] -> A
    U42 :: [A] -> A
    U51 :: [A x A] -> A
    U52 :: [A] -> A
    U61 :: [A x A] -> A
    U62 :: [A] -> A
    U71 :: [A x A x A] -> A
    U72 :: [A x A] -> A
    s :: [A] -> A
    length :: [A] -> A
    U81 :: [A] -> A
    U91 :: [A x A x A x A] -> A
    U92 :: [A x A x A x A] -> A
    U93 :: [A x A x A x A] -> A
    take :: [A x A] -> A
    proper :: [A] -> A
    isNatIList :: [A] -> A
    isNatList :: [A] -> A
    isNat :: [A] -> A
    top :: [A] -> A)
(RULES
    active(zeros()) -> mark(cons(0()
                                ,zeros()))
    active(U11(tt())) -> mark(tt())
    active(U21(tt())) -> mark(tt())
    active(U31(tt())) -> mark(tt())
    active(U41(tt(),V2)) ->
      mark(U42(isNatIList(V2)))
    active(U42(tt())) -> mark(tt())
    active(U51(tt(),V2)) ->
      mark(U52(isNatList(V2)))
    active(U52(tt())) -> mark(tt())
    active(U61(tt(),V2)) ->
      mark(U62(isNatIList(V2)))
    active(U62(tt())) -> mark(tt())
    active(U71(tt(),L,N)) ->
      mark(U72(isNat(N),L))
    active(U72(tt(),L)) ->
      mark(s(length(L)))
    active(U81(tt())) -> mark(nil())
    active(U91(tt(),IL,M,N)) ->
      mark(U92(isNat(M),IL,M,N))
    active(U92(tt(),IL,M,N)) ->
      mark(U93(isNat(N),IL,M,N))
    active(U93(tt(),IL,M,N)) ->
      mark(cons(N,take(M,IL)))
    active(isNat(0())) -> mark(tt())
    active(isNat(length(V1))) ->
      mark(U11(isNatList(V1)))
    active(isNat(s(V1))) ->
      mark(U21(isNat(V1)))
    active(isNatIList(V)) ->
      mark(U31(isNatList(V)))
    active(isNatIList(zeros())) ->
      mark(tt())
    active(isNatIList(cons(V1
                          ,V2))) -> mark(U41(isNat(V1)
                                            ,V2))
    active(isNatList(nil())) ->
      mark(tt())
    active(isNatList(cons(V1
                         ,V2))) -> mark(U51(isNat(V1)
                                           ,V2))
    active(isNatList(take(V1
                         ,V2))) -> mark(U61(isNat(V1)
                                           ,V2))
    active(length(nil())) ->
      mark(0())
    active(length(cons(N,L))) ->
      mark(U71(isNatList(L),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)))