Write You a Haskell Stephen Diehl

λx.e
ere are several syntactical conventions that we will adopt when writing lambda expressions. Application
of multiple expressions associates to the left.
x 1 x 2 x 3 ...x n = (...((x 1 x 2 )x 3 )...x n )
By convention application extends as far to the right as is syntactically meaningful. Parenthesis are used
to disambiguate.
In the lambda calculus all lambda abstractions bind a single variable, their body may be another lambda
abstraction. Out of convenience we often write multiple lambda abstractions with their variables on one
lambda symbol. is is merely a syntactical convention and does not change the underlying meaning.
λxy.z = λx.λy.z
e actual implementation of the lambda calculus admits several degrees of freedom in how they are
represented. e most notable is the choice of identifier for the binding variables. A variable is said to
be bound if it is contained in a lambda expression of the same variable binding. Conversely a variable is
free if it is not bound.
A term with free variables is said to be an open term while one without free variables is said to be closed
or a combinator.
e 0 = λx.x
e 1 = λx.(x(λy.ya)x)y
e 0 is a combinator while e 1 is not. In e 1 both occurances of x are bound. e first y is bound, while the
second is free. a is also free.
Multiple lambda abstractions may bind the same variable name. Each occurance of a variable is then
bound by the nearest enclosing binder. For example the x variable in the following expression is bound
on the inner lambda, while y is bound on the outer lambda. is phenomenon is referred to as name
shadowing.
λxy.(λxz.x + y)
SKI Combinators
ere are three fundamental closed expressions called the SKI combinators.
S = λf.(λg.(λx.fx(gx)))
K = λx.λy.x
I = λx.x
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