Type rules

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 program is well typed and what type expressions 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.

An expression ${\displaystyle e}$ of type ${\displaystyle \tau }$ is written as ${\displaystyle e\!:\!\tau }$. The typing environment is written as ${\displaystyle \Gamma }$. The notation for inference is the usual one for sequents and inference rules, and has the following general form

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