function type (changes)

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Let $A,B$ be two types. We introduce a type $A \to B$ called the **function type**. The constructors for this type are written as

$(\lambda x . y) : A \to B$

where $x:A$ and $y:B$.

These can be computed (on application) using $\beta$-reduction:

$$(\lambda x.y)\mathrm{za}\equiv y[\mathrm{xa}/\mathrm{zx}]$$ (\lambda x.~~ y)z~~ y)a \equiv~~ y[x/z]~~ y[a/x]

which says take $y$ and replace all occurrences of $x$ with $\mathrm{za}$~~ z~~ a for an $a : A$.

Given any type $A$, there is a function called the *identity function* of $A$ denoted and defined by

$id_A : A \to A$

$id_A \equiv \lambda x.x$

Given any types $A,B,C$ and two functions $f : A \to B$, $g : B \to C$ the composition of $f$ and $g$ is the function

$g \circ f : A \to C$

$(g \circ f) (x) \equiv g(f(x))$

category: type theory

Last revised on January 19, 2019 at 10:22:54. See the history of this page for a list of all contributions to it.