(STRATEGY
    INNERMOST)

(VAR
    M N V1 V2 X X1 X2)
(DATATYPES
    A = µX.< tt, n__0, n__plus(X, X), n__isNatKind(X), n__s(X), n__and(X, X), n__isNat(X) >)
(SIGNATURES
    U11 :: [A x A x A] -> A
    U12 :: [A x A] -> A
    U13 :: [A] -> A
    U21 :: [A x A] -> A
    U22 :: [A] -> A
    U31 :: [A x A] -> A
    U41 :: [A x A x A] -> A
    and :: [A x A] -> A
    isNat :: [A] -> A
    isNatKind :: [A] -> A
    plus :: [A x A] -> A
    0 :: [] -> A
    s :: [A] -> A
    activate :: [A] -> A)
(RULES
    U11(tt(),V1,V2) ->
      U12(isNat(activate(V1))
         ,activate(V2))
    U12(tt(),V2) ->
      U13(isNat(activate(V2)))
    U13(tt()) -> tt()
    U21(tt(),V1) ->
      U22(isNat(activate(V1)))
    U22(tt()) -> tt()
    U31(tt(),N) -> activate(N)
    U41(tt(),M,N) ->
      s(plus(activate(N),activate(M)))
    and(tt(),X) -> activate(X)
    isNat(n__0()) -> tt()
    isNat(n__plus(V1,V2)) ->
      U11(and(isNatKind(activate(V1))
             ,n__isNatKind(activate(V2)))
         ,activate(V1)
         ,activate(V2))
    isNat(n__s(V1)) ->
      U21(isNatKind(activate(V1))
         ,activate(V1))
    isNatKind(n__0()) -> tt()
    isNatKind(n__plus(V1,V2)) ->
      and(isNatKind(activate(V1))
         ,n__isNatKind(activate(V2)))
    isNatKind(n__s(V1)) ->
      isNatKind(activate(V1))
    plus(N,0()) -> U31(and(isNat(N)
                          ,n__isNatKind(N))
                      ,N)
    plus(N,s(M)) ->
      U41(and(and(isNat(M)
                 ,n__isNatKind(M))
             ,n__and(n__isNat(N)
                    ,n__isNatKind(N)))
         ,M
         ,N)
    0() -> n__0()
    plus(X1,X2) -> n__plus(X1,X2)
    isNatKind(X) -> n__isNatKind(X)
    s(X) -> n__s(X)
    and(X1,X2) -> n__and(X1,X2)
    isNat(X) -> n__isNat(X)
    activate(n__0()) -> 0()
    activate(n__plus(X1,X2)) ->
      plus(activate(X1),activate(X2))
    activate(n__isNatKind(X)) ->
      isNatKind(X)
    activate(n__s(X)) ->
      s(activate(X))
    activate(n__and(X1,X2)) ->
      and(activate(X1),X2)
    activate(n__isNat(X)) ->
      isNat(X)
    activate(X) -> X)