Typing environment information

In type theory a typing environment (or typing context) represents the association between variable names and data types.

More formally an environment $\Gamma$ is a set or ordered list of pairs $\langle x,\tau \rangle$ , usually written as $x:\tau$ , where $x$ is a variable and $\tau$ its type.

The judgement

\Gamma \vdash e:\tau

is read as " $e$ has type $\tau$ in context $\Gamma$ ".^[1]

For each function body type checks:

\Gamma =\{(f,\tau _{1}\times ...\times \tau _{n}\to \tau _{0})|(f,xs,(\tau _{1},...,\tau _{n}),t_{f},\tau _{0})\in e\}

Typing Rules Example:

{\begin{array}{c}\Gamma \vdash b:Bool,\Gamma \vdash t_{1}:\tau ,\Gamma \vdash t_{2}:\tau \\\hline \Gamma \vdash ({\text{if}}(b)t_{1}{\text{else}}t_{2}):\tau \\\end{array}}

In statically typed programming languages these environments are used and maintained by typing rules to type check a given program or expression.

^ "Simply Typed λ-calculus" (PDF).

[cornell-cs4110-1] "Simply Typed λ-calculus" (PDF).

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