(STRATEGY
    INNERMOST)

(VAR
    M N V1 V2 X X1 X2)
(DATATYPES
    A = µX.< tt, n__0, n__plus(X, X), n__s(X), n__x(X, X) >)
(SIGNATURES
    U101 :: [A x A x A] -> A
    U102 :: [A x A x A] -> A
    U103 :: [A x A x A] -> A
    U104 :: [A x A x A] -> A
    U11 :: [A x A x A] -> A
    U12 :: [A x A x A] -> A
    U13 :: [A x A x A] -> A
    U14 :: [A x A x A] -> A
    U15 :: [A x A] -> A
    U16 :: [A] -> A
    U21 :: [A x A] -> A
    U22 :: [A x A] -> A
    U23 :: [A] -> A
    U31 :: [A x A x A] -> A
    U32 :: [A x A x A] -> A
    U33 :: [A x A x A] -> A
    U34 :: [A x A x A] -> A
    U35 :: [A x A] -> A
    U36 :: [A] -> A
    U41 :: [A x A] -> A
    U42 :: [A] -> A
    U51 :: [A] -> A
    U61 :: [A x A] -> A
    U62 :: [A] -> A
    U71 :: [A x A] -> A
    U72 :: [A x A] -> A
    U81 :: [A x A x A] -> A
    U82 :: [A x A x A] -> A
    U83 :: [A x A x A] -> A
    U84 :: [A x A x A] -> A
    U91 :: [A x A] -> A
    U92 :: [A] -> A
    isNat :: [A] -> A
    isNatKind :: [A] -> A
    plus :: [A x A] -> A
    x :: [A x A] -> A
    0 :: [] -> A
    s :: [A] -> A
    activate :: [A] -> A)
(RULES
    U101(tt(),M,N) ->
      U102(isNatKind(activate(M))
          ,activate(M)
          ,activate(N))
    U102(tt(),M,N) ->
      U103(isNat(activate(N))
          ,activate(M)
          ,activate(N))
    U103(tt(),M,N) ->
      U104(isNatKind(activate(N))
          ,activate(M)
          ,activate(N))
    U104(tt(),M,N) ->
      plus(x(activate(N),activate(M))
          ,activate(N))
    U11(tt(),V1,V2) ->
      U12(isNatKind(activate(V1))
         ,activate(V1)
         ,activate(V2))
    U12(tt(),V1,V2) ->
      U13(isNatKind(activate(V2))
         ,activate(V1)
         ,activate(V2))
    U13(tt(),V1,V2) ->
      U14(isNatKind(activate(V2))
         ,activate(V1)
         ,activate(V2))
    U14(tt(),V1,V2) ->
      U15(isNat(activate(V1))
         ,activate(V2))
    U15(tt(),V2) ->
      U16(isNat(activate(V2)))
    U16(tt()) -> tt()
    U21(tt(),V1) ->
      U22(isNatKind(activate(V1))
         ,activate(V1))
    U22(tt(),V1) ->
      U23(isNat(activate(V1)))
    U23(tt()) -> tt()
    U31(tt(),V1,V2) ->
      U32(isNatKind(activate(V1))
         ,activate(V1)
         ,activate(V2))
    U32(tt(),V1,V2) ->
      U33(isNatKind(activate(V2))
         ,activate(V1)
         ,activate(V2))
    U33(tt(),V1,V2) ->
      U34(isNatKind(activate(V2))
         ,activate(V1)
         ,activate(V2))
    U34(tt(),V1,V2) ->
      U35(isNat(activate(V1))
         ,activate(V2))
    U35(tt(),V2) ->
      U36(isNat(activate(V2)))
    U36(tt()) -> tt()
    U41(tt(),V2) ->
      U42(isNatKind(activate(V2)))
    U42(tt()) -> tt()
    U51(tt()) -> tt()
    U61(tt(),V2) ->
      U62(isNatKind(activate(V2)))
    U62(tt()) -> tt()
    U71(tt(),N) ->
      U72(isNatKind(activate(N))
         ,activate(N))
    U72(tt(),N) -> activate(N)
    U81(tt(),M,N) ->
      U82(isNatKind(activate(M))
         ,activate(M)
         ,activate(N))
    U82(tt(),M,N) ->
      U83(isNat(activate(N))
         ,activate(M)
         ,activate(N))
    U83(tt(),M,N) ->
      U84(isNatKind(activate(N))
         ,activate(M)
         ,activate(N))
    U84(tt(),M,N) ->
      s(plus(activate(N),activate(M)))
    U91(tt(),N) ->
      U92(isNatKind(activate(N)))
    U92(tt()) -> 0()
    isNat(n__0()) -> tt()
    isNat(n__plus(V1,V2)) ->
      U11(isNatKind(activate(V1))
         ,activate(V1)
         ,activate(V2))
    isNat(n__s(V1)) ->
      U21(isNatKind(activate(V1))
         ,activate(V1))
    isNat(n__x(V1,V2)) ->
      U31(isNatKind(activate(V1))
         ,activate(V1)
         ,activate(V2))
    isNatKind(n__0()) -> tt()
    isNatKind(n__plus(V1,V2)) ->
      U41(isNatKind(activate(V1))
         ,activate(V2))
    isNatKind(n__s(V1)) ->
      U51(isNatKind(activate(V1)))
    isNatKind(n__x(V1,V2)) ->
      U61(isNatKind(activate(V1))
         ,activate(V2))
    plus(N,0()) -> U71(isNat(N),N)
    plus(N,s(M)) -> U81(isNat(M)
                       ,M
                       ,N)
    x(N,0()) -> U91(isNat(N),N)
    x(N,s(M)) -> U101(isNat(M),M,N)
    0() -> n__0()
    plus(X1,X2) -> n__plus(X1,X2)
    s(X) -> n__s(X)
    x(X1,X2) -> n__x(X1,X2)
    activate(n__0()) -> 0()
    activate(n__plus(X1,X2)) ->
      plus(X1,X2)
    activate(n__s(X)) -> s(X)
    activate(n__x(X1,X2)) -> x(X1
                              ,X2)
    activate(X) -> X)