Simplifying Lambda Syntax

Evaluating lambda functions requires using lots of brackets, which can be tedious and are a major source of error for evaluating expressions by hand. To simplify expressions, you can omit brackets when it is clear what the intention of the function is. Particularly, a function application can omit the brackets surrounding each individual parameter and assume the function is applied to the nearest argument. So, instead of expressing a function of three arguments as

$$ (((f \: a1) \: a2) \: a3) $$

We can write

$$ f \: a1 \: a2 \: a3 $$

A second simplification of notation is defining functions. Instead of defining a function using the full lambda syntax already described, we can drop the $\lambda$ symbol and move any variables of the function definition to the left of the $=$ sign. This means a function definition like

$$ def \: f = \lambda\langle name \rangle. \langle expression \rangle $$

can be expressed as

$$ def \: f \langle name \rangle = \langle expression \rangle $$

Using this syntactic sugar, we can express common functions like identity in a simplified format.

$$ def \: identity = \lambda x.x $$


$$ def \: identity \: x = x $$

You will see this simplified syntax used to express lambda expressions more succinctly, especially when we start building more complex lambda expressions that rely on conditional logic or recursion.

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