(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
    U11 :: [A x A] -> A
    U12 :: [A] -> A
    U21 :: [A] -> A
    U31 :: [A x A] -> A
    U32 :: [A] -> A
    U41 :: [A x A] -> A
    U51 :: [A x A x A] -> A
    U52 :: [A x A x A] -> A
    U61 :: [A] -> A
    U71 :: [A x A x A] -> A
    U72 :: [A x A x A] -> A
    isNat :: [A] -> A
    plus :: [A x A] -> A
    x :: [A x A] -> A
    0 :: [] -> A
    s :: [A] -> A
    activate :: [A] -> A)
(RULES
    U11(tt(),V2) ->
      U12(isNat(activate(V2)))
    U12(tt()) -> tt()
    U21(tt()) -> tt()
    U31(tt(),V2) ->
      U32(isNat(activate(V2)))
    U32(tt()) -> tt()
    U41(tt(),N) -> activate(N)
    U51(tt(),M,N) ->
      U52(isNat(activate(N))
         ,activate(M)
         ,activate(N))
    U52(tt(),M,N) ->
      s(plus(activate(N),activate(M)))
    U61(tt()) -> 0()
    U71(tt(),M,N) ->
      U72(isNat(activate(N))
         ,activate(M)
         ,activate(N))
    U72(tt(),M,N) ->
      plus(x(activate(N),activate(M))
          ,activate(N))
    isNat(n__0()) -> tt()
    isNat(n__plus(V1,V2)) ->
      U11(isNat(activate(V1))
         ,activate(V2))
    isNat(n__s(V1)) ->
      U21(isNat(activate(V1)))
    isNat(n__x(V1,V2)) ->
      U31(isNat(activate(V1))
         ,activate(V2))
    plus(N,0()) -> U41(isNat(N),N)
    plus(N,s(M)) -> U51(isNat(M)
                       ,M
                       ,N)
    x(N,0()) -> U61(isNat(N))
    x(N,s(M)) -> U71(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(activate(X1),activate(X2))
    activate(n__s(X)) ->
      s(activate(X))
    activate(n__x(X1,X2)) ->
      x(activate(X1),activate(X2))
    activate(X) -> X)