Write You a Haskell Stephen Diehl

Eta-expansion
(λx.ex)e ′ β
→ ee
′
if x /∈ fv(e)
e opposite of eta reduction is eta-expansion, which takes a function that is not saturated and makes
all variables explicitly bound in a lambda. Eta-expansion will be important when we discuss translation
into STG.
Reduction
Evaluation of lambda calculus expressions proceeds by beta reduction. e variables bound in a lambda
are substituted across the body of the lambda. ere are several degrees of freedom in the design space
about how to do this, and in which order an expression should be evaluated. For instance we could
evaluate under the lambda and then substitute variables into it, or instead evaluate the arguments and
then substitute and then reduce the lambda expressions. More on this will be discussed in the section
on Evaluation models.
Untyped> (\x.x) 1
1
Untyped> (\x y . y) 1 2
2
Untyped> (\x y z. x z (y z)) (\x y . x) (\x y . x)
=> \x y z . (x z (y z))
=> \y z . ((\x y . x) z (y z))
=> \x y . x
=> \y . z
=> z
=> \z . z
\z . z
In the untyped lambda calculus we can freely represent infinitely diverging expressions:
Untyped> \f . (f (\x . (f x x)) (\x . (f x x)))
\f . (f (\x . (f x x)) (\x . (f x x)))
Untyped> (\f . (\x. (f x x)) (\x. (f x x))) (\f x . f f)
...
Untyped> (\x. x x) (\x. x x)
...
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