Write You a Haskell Stephen Diehl

e fundamental issue with using locally named binders is the problem of name capture, or how to handle
the case where a substitution conflicts with the names of free variables. We need the condition in the
last case to avoid the naive substitution that would fundamentally alter the meaning of the following
expression when y is rewritten to x.
[y/x](λx.xy) → λx.xx
By convention we will always use a capture-avoiding substitution. Substitution will only proceed if the
variable is not in the set of free variables of the expression, and if it does then a fresh variable will be
created in its place.
(λx.e)a → [x/a]e
if x /∈ fv(a)
ere are several binding libraries and alternative implementations of the lambda calculus syntax that
avoid these problems. It is a very common problem and it is very easy to implement incorrectly even for
experts.
Conversion and Equivalences
Alpha equivalence
(λx.e) α = (λy.[x/y]e)
Alpha equivalence is the property ( when using named binders ) that changing the variable on the binder
and throughout the body of the expression should not change the fundamental meaning of the whole
expression. So for example the following are alpha-equivalent.
λxy.xy
α
= λab.ab
Beta-reduction
Beta reduction is simply a single substitution step, replacing a variable bound by a lambda expression
with the argument to the lambda throughout the body of the expression.
Eta-reduction
(λx.a)y β → [x/y]a
λx.ex η → e if x /∈ fv(e)
is is justified by the fact that if we apply both sides to a term, one step of beta reduction turns the left
side to the right side:
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