The syntax of the λ-calculus is very simple, comprising
just four kinds of expression but surprisingly
it is sufficient to define any computable function.

Constants including
integers, booleans, conditiional, declarations
and so on can all be defined using just the syntax above,
as demonstrated by examples.
Because of this it is sometimes convenient to extend the
syntax with options for constants
(0, 1, 2, ..., +, -, *, /,
=, ≠, ≤, <, ≥, >,
true, false, ∧, ∨, ¬,
nil, cons, null, hd, tl,
if...then...else..., let...in... etc.) but,
if that is done, it is only a convenience and does not
increase the power of the language.

Note that an abstraction defines an anonymous function.

The pure λ calculus appears to lack recursion
(or equivalently iteration)
but recursive functions can in fact be defined,
as also demonstrated by examples.

let rec fact = lambda n . if n <= 0 then 1 else n*fact(n-1)
in fact 10

output area

This prototype javascript
implementation of λ (22/8/'17)
will eventually spread to other pages.

Please turn Javascript on.

The introduction describes
the semantics of λ calculus and programming techniques, and
an interpreter shows how it can be made to work.