Recursive equations vs. inference rules It seems to me that recursive equations can always be presented as inference rules.  For the forward direction, an example is addition over Peono numerals (built from $O$ and $S(\_)$)

$$
\begin{align}
add(O, m)    &amp; = m \\
add(S(n), m) &amp; = S(add(n, m))
\end{align}
$$

which can be also presented as

$$
\frac{ }{add(O, m) = m} \qquad \frac{add(n, m) = a}{add(S(n), m) = S(a)}
$$

Is the backward conversion (from inference rules to recursive equations) always possible?  If not, it seems inference rules more general, right?  Except the superficial difference between the presentations, is there any other deeper difference between the two?
Inference rules are "more general". A recursive equation, actually any equation $E_1 = E_2$ is like an inference rule without premisses:
$$\frac{}{E_1 = E_2}$$
A general inference rule may have premisses, and so it is like a conditional equation.
Only very special kinds of inference rules may be equivalently expressed as equations. For instance, it seems difficult to express
$$\frac{a \bullet c = b \bullet c}{a = b}$$
and
$$\frac{add(x,add(y,z)) = add(x,z)}{add(y,z) = z}$$
as equations.