(STRATEGY
    INNERMOST)

(VAR
    M N V1 V2 X X1 X2 X3)
(DATATYPES
    A = µX.< tt, s(X), 0, plus(X, X), isNatKind(X), and(X, X), isNat(X), U11(X, X, X), U12(X, X), U13(X), U21(X, X), U22(X), U31(X, X), U41(X, X, X) >)
(SIGNATURES
    a__U11 :: [A x A x A] -> A
    a__U12 :: [A x A] -> A
    a__U13 :: [A] -> A
    a__U21 :: [A x A] -> A
    a__U22 :: [A] -> A
    a__U31 :: [A x A] -> A
    a__U41 :: [A x A x A] -> A
    a__and :: [A x A] -> A
    a__isNat :: [A] -> A
    a__isNatKind :: [A] -> A
    a__plus :: [A x A] -> A
    mark :: [A] -> A)
(RULES
    a__U11(tt(),V1,V2) ->
      a__U12(a__isNat(V1),V2)
    a__U12(tt(),V2) ->
      a__U13(a__isNat(V2))
    a__U13(tt()) -> tt()
    a__U21(tt(),V1) ->
      a__U22(a__isNat(V1))
    a__U22(tt()) -> tt()
    a__U31(tt(),N) -> mark(N)
    a__U41(tt(),M,N) ->
      s(a__plus(mark(N),mark(M)))
    a__and(tt(),X) -> mark(X)
    a__isNat(0()) -> tt()
    a__isNat(plus(V1,V2)) ->
      a__U11(a__and(a__isNatKind(V1)
                   ,isNatKind(V2))
            ,V1
            ,V2)
    a__isNat(s(V1)) ->
      a__U21(a__isNatKind(V1),V1)
    a__isNatKind(0()) -> tt()
    a__isNatKind(plus(V1,V2)) ->
      a__and(a__isNatKind(V1)
            ,isNatKind(V2))
    a__isNatKind(s(V1)) ->
      a__isNatKind(V1)
    a__plus(N,0()) ->
      a__U31(a__and(a__isNat(N)
                   ,isNatKind(N))
            ,N)
    a__plus(N,s(M)) ->
      a__U41(a__and(a__and(a__isNat(M)
                          ,isNatKind(M))
                   ,and(isNat(N),isNatKind(N)))
            ,M
            ,N)
    mark(U11(X1,X2,X3)) ->
      a__U11(mark(X1),X2,X3)
    mark(U12(X1,X2)) ->
      a__U12(mark(X1),X2)
    mark(isNat(X)) -> a__isNat(X)
    mark(U13(X)) -> a__U13(mark(X))
    mark(U21(X1,X2)) ->
      a__U21(mark(X1),X2)
    mark(U22(X)) -> a__U22(mark(X))
    mark(U31(X1,X2)) ->
      a__U31(mark(X1),X2)
    mark(U41(X1,X2,X3)) ->
      a__U41(mark(X1),X2,X3)
    mark(plus(X1,X2)) ->
      a__plus(mark(X1),mark(X2))
    mark(and(X1,X2)) ->
      a__and(mark(X1),X2)
    mark(isNatKind(X)) ->
      a__isNatKind(X)
    mark(tt()) -> tt()
    mark(s(X)) -> s(mark(X))
    mark(0()) -> 0()
    a__U11(X1,X2,X3) -> U11(X1
                           ,X2
                           ,X3)
    a__U12(X1,X2) -> U12(X1,X2)
    a__isNat(X) -> isNat(X)
    a__U13(X) -> U13(X)
    a__U21(X1,X2) -> U21(X1,X2)
    a__U22(X) -> U22(X)
    a__U31(X1,X2) -> U31(X1,X2)
    a__U41(X1,X2,X3) -> U41(X1
                           ,X2
                           ,X3)
    a__plus(X1,X2) -> plus(X1,X2)
    a__and(X1,X2) -> and(X1,X2)
    a__isNatKind(X) -> isNatKind(X))