(STRATEGY
    INNERMOST)

(VAR
    IL L M N V V1 V2 X X1 X2)
(DATATYPES
    A = µX.< n__zeros, tt, n__take(X, X), n__0, n__length(X), n__s(X), n__cons(X, X), n__isNatIListKind(X), n__nil, n__and(X, X), n__isNat(X), n__isNatKind(X) >)
(SIGNATURES
    zeros :: [] -> A
    U11 :: [A x A] -> A
    U12 :: [A] -> A
    U21 :: [A x A] -> A
    U22 :: [A] -> A
    U31 :: [A x A] -> A
    U32 :: [A] -> A
    U41 :: [A x A x A] -> A
    U42 :: [A x A] -> A
    U43 :: [A] -> A
    U51 :: [A x A x A] -> A
    U52 :: [A x A] -> A
    U53 :: [A] -> A
    U61 :: [A x A x A] -> A
    U62 :: [A x A] -> A
    U63 :: [A] -> A
    U71 :: [A x A] -> A
    U81 :: [A] -> A
    U91 :: [A x A x A x A] -> A
    and :: [A x A] -> A
    isNat :: [A] -> A
    isNatIList :: [A] -> A
    isNatIListKind :: [A] -> A
    isNatKind :: [A] -> A
    isNatList :: [A] -> A
    length :: [A] -> A
    take :: [A x A] -> A
    0 :: [] -> A
    s :: [A] -> A
    cons :: [A x A] -> A
    nil :: [] -> A
    activate :: [A] -> A)
(RULES
    zeros() -> cons(0(),n__zeros())
    U11(tt(),V1) ->
      U12(isNatList(activate(V1)))
    U12(tt()) -> tt()
    U21(tt(),V1) ->
      U22(isNat(activate(V1)))
    U22(tt()) -> tt()
    U31(tt(),V) ->
      U32(isNatList(activate(V)))
    U32(tt()) -> tt()
    U41(tt(),V1,V2) ->
      U42(isNat(activate(V1))
         ,activate(V2))
    U42(tt(),V2) ->
      U43(isNatIList(activate(V2)))
    U43(tt()) -> tt()
    U51(tt(),V1,V2) ->
      U52(isNat(activate(V1))
         ,activate(V2))
    U52(tt(),V2) ->
      U53(isNatList(activate(V2)))
    U53(tt()) -> tt()
    U61(tt(),V1,V2) ->
      U62(isNat(activate(V1))
         ,activate(V2))
    U62(tt(),V2) ->
      U63(isNatIList(activate(V2)))
    U63(tt()) -> tt()
    U71(tt(),L) ->
      s(length(activate(L)))
    U81(tt()) -> nil()
    U91(tt(),IL,M,N) ->
      cons(activate(N)
          ,n__take(activate(M)
                  ,activate(IL)))
    and(tt(),X) -> activate(X)
    isNat(n__0()) -> tt()
    isNat(n__length(V1)) ->
      U11(isNatIListKind(activate(V1))
         ,activate(V1))
    isNat(n__s(V1)) ->
      U21(isNatKind(activate(V1))
         ,activate(V1))
    isNatIList(V) ->
      U31(isNatIListKind(activate(V))
         ,activate(V))
    isNatIList(n__zeros()) -> tt()
    isNatIList(n__cons(V1,V2)) ->
      U41(and(isNatKind(activate(V1))
             ,n__isNatIListKind(activate(V2)))
         ,activate(V1)
         ,activate(V2))
    isNatIListKind(n__nil()) -> tt()
    isNatIListKind(n__zeros()) ->
      tt()
    isNatIListKind(n__cons(V1
                          ,V2)) ->
      and(isNatKind(activate(V1))
         ,n__isNatIListKind(activate(V2)))
    isNatIListKind(n__take(V1
                          ,V2)) ->
      and(isNatKind(activate(V1))
         ,n__isNatIListKind(activate(V2)))
    isNatKind(n__0()) -> tt()
    isNatKind(n__length(V1)) ->
      isNatIListKind(activate(V1))
    isNatKind(n__s(V1)) ->
      isNatKind(activate(V1))
    isNatList(n__nil()) -> tt()
    isNatList(n__cons(V1,V2)) ->
      U51(and(isNatKind(activate(V1))
             ,n__isNatIListKind(activate(V2)))
         ,activate(V1)
         ,activate(V2))
    isNatList(n__take(V1,V2)) ->
      U61(and(isNatKind(activate(V1))
             ,n__isNatIListKind(activate(V2)))
         ,activate(V1)
         ,activate(V2))
    length(nil()) -> 0()
    length(cons(N,L)) ->
      U71(and(and(isNatList(activate(L))
                 ,n__isNatIListKind(activate(L)))
             ,n__and(n__isNat(N)
                    ,n__isNatKind(N)))
         ,activate(L))
    take(0(),IL) ->
      U81(and(isNatIList(IL)
             ,n__isNatIListKind(IL)))
    take(s(M),cons(N,IL)) ->
      U91(and(and(isNatIList(activate(IL))
                 ,n__isNatIListKind(activate(IL)))
             ,n__and(n__and(n__isNat(M)
                           ,n__isNatKind(M))
                    ,n__and(n__isNat(N)
                           ,n__isNatKind(N))))
         ,activate(IL)
         ,M
         ,N)
    zeros() -> n__zeros()
    take(X1,X2) -> n__take(X1,X2)
    0() -> n__0()
    length(X) -> n__length(X)
    s(X) -> n__s(X)
    cons(X1,X2) -> n__cons(X1,X2)
    isNatIListKind(X) ->
      n__isNatIListKind(X)
    nil() -> n__nil()
    and(X1,X2) -> n__and(X1,X2)
    isNat(X) -> n__isNat(X)
    isNatKind(X) -> n__isNatKind(X)
    activate(n__zeros()) -> zeros()
    activate(n__take(X1,X2)) ->
      take(activate(X1),activate(X2))
    activate(n__0()) -> 0()
    activate(n__length(X)) ->
      length(activate(X))
    activate(n__s(X)) ->
      s(activate(X))
    activate(n__cons(X1,X2)) ->
      cons(activate(X1),X2)
    activate(n__isNatIListKind(X)) ->
      isNatIListKind(X)
    activate(n__nil()) -> nil()
    activate(n__and(X1,X2)) ->
      and(activate(X1),X2)
    activate(n__isNat(X)) ->
      isNat(X)
    activate(n__isNatKind(X)) ->
      isNatKind(X)
    activate(X) -> X)