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In [[type theory]], a '''type rule''' is an [[inference rule]] that describes how a [[type system]] assigns a type to a syntactic construction. These rules may be applied by the type system to determine if a [[Computer program|program]] is well typed and what type [[Expression (computer science)|expression]]s have. A prototypical example of the use of type rules is in defining type inference in the [[simply typed lambda calculus]], which is the [[internal language]] of [[Cartesian closed categories]].
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== Notation ==
<!-- historical reference (what it the paper by Luca Cardelli that introduce this notation?) -->
An expression <math>e</math> of type <math>\tau</math> is written as <math>e\!:\!\tau</math>. The [[typing environment]] is written as <math>\Gamma</math>. The notation for inference is the usual one for [[sequent]]s and [[inference rule]]s, and has the following general form
 
:<math>
\frac{\Gamma_1 \vdash e_1\!:\!\tau_1 \quad \cdots \quad \Gamma_n \vdash e_n\!:\!\tau_n}{\Gamma \vdash e\!:\!\tau}
</math>
 
The sequents above the line are the premises that must be fulfilled for the rule to be applied, yielding the conclusion: the sequents below the line. This can be read as: ''if expression <math>e_i</math> has type <math>\tau_i</math> in [[Typing environment|environment]] <math>\Gamma_i</math>, for all <math>i=1..n</math>, then the expression <math>e</math> will have an environment <math>\Gamma</math> and type <math>\tau</math>''.
 
For example, a simple language to perform arithmetic calculations on real numbers may have the following rules
 
:<math>
\frac{\Gamma \vdash e_1\!:\!real \quad \Gamma \vdash e_2\!:\!real}{\Gamma \vdash e_1+e_2\!:\!real}
\qquad \frac{\Gamma \vdash e_1\!:\!integer \quad \Gamma \vdash e_2 : integer}{\Gamma \vdash e_1+e_2\!:\!integer} \qquad \cdots
</math>
 
A type rule may have no premises, and usually the line is omitted in these cases. A type rule may also change an environment by adding new variables to a previous environment; for example, a declaration may have the following type rule, where a new variable <math>id</math>,
with type <math>\tau'</math>, is added to <math>\Gamma</math>:
 
:<math>
\frac{\Gamma \vdash e'\!:\!\tau' \quad \Gamma, id\!:\!\tau' \vdash e\!:\!\tau}{\Gamma \vdash \text{let id = } e' \text{ in } e \text{ end} :\!\tau}
</math>
 
This types can be used to derive composed expressions types, much like in [[natural deduction]].
 
==See also==
* [[Judgment (mathematical logic)]]
* [[Type system]]
* [[Type theory]]
* [[Curry–Howard correspondence]]
 
== Further reading ==
*  Luca Cardelli, "Type Systems", ''ACM Computing Surveys''
 
{{type-theory-stub}}
 
[[Category:Data types]]
[[Category:Program analysis]]
[[Category:Type theory]]

Revision as of 19:52, 7 February 2014

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