Some authors reserve the term "term" for just variables and abstracts. We'll probably just say "term" and "expression" indiscriminately for expressions of any of these three forms.
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Examples of expressions:
x
(x (\x x))
((\x (x x)) (\x (x x)))
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The lambda calculus has an associated proof theory. For now, we can regard the
proof theory as having just one rule, called the rule of **beta-reduction** or
"beta-contraction". Suppose you have some expression of the form:
It's easy to be lulled into thinking this is a kind of imperative construction. *But it's not!* It's really just a shorthand for the compound "let"-expressions we've already been looking at, taking the maximum syntactically permissible scope. (Compare the "dot" convention in the lambda calculus, discussed above.)
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9. Some shorthand
OCaml permits you to abbreviate:
or in other words, interpret the rest of the file or interactive session with `bar` assigned the function `(lambda (x) B)`.
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10. Shadowing
When a previously-bound variable is rebound in the way we see here, that's called **shadowing**: the outer binding is shadowed during the scope of the inner binding.
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Some more comparisons between Scheme and OCaml
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