It is well known that the full reduction tree for any term of the typed 2-calculus is finite. However, it is not obvious how a reasonable estimate for its height might be obtained. Here we note that the head reduction tree has the property that the number of its nodes with conversions bounds the length of any reduction sequence*. The height of that tree, and hence also the number of its nodes, can be estimated using a technique due to Howard [31 which in turn is based on work of Sanchis [4] and

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... Diller [11. This gives the desired upper bound. The method of Gandy [2] can also be used to obtain a bound for the length of arbitrary reduction sequences; this is carried out in [5] . However, the bound derived here, apart from being more intelligible, is also better. Let r, s, t denote terms of the typed 2-calculus. The level lev(r) of r is defined to be the level lev(Q) of its type Q, where ground types have level 0 and lev(Q~a) =max(lev(Q) + 1, lev(a)). For r of level 0 we define ~r inductively by ~-Rule. If a -~rx[s]t, then I~, + l(~.xr)st. a ~ a+l -Variable Rule. If Fmtiyi lor i = 1 .... , n, then m xtl ... t,. In particular, ~t~---x' a. 1 for any a and m. -Cut Rule. If ~mryl... y, with n > 1 and ~ti3~ and lev(ti) < m for i = 1,..., n, then a+l [ m rt l "" tn. Note that ~r is generated by a uniquely determined rule. Hence the generation tree (with the a's stripped off) is uniquely determined; we call it the head reduction tree of r. Variable Lemma. If lev(x) < a, then ~mX~ Proof. By induction on lev(x). By induction hypothesis I ma-1 yiz~,+ hence ~mxya ~ by the Variable Rule. [] * This is not quite true, but only for so-called 2-/-terms, where any variable bound by ,t actually occurs in the kernel. But the general case can be easily reduced to this one by introducing dummy variables; this is carried out below