In type theory, a typing 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 typing rules is in defining type inference in the simply typed lambda calculus, which is the internal language of Cartesian closed categories. Typing rules specify the structure of a typing relation that relates syntactic terms to their types. Syntactically, the typing relation is usually denoted by a colon, so for example denotes that an expression has type . The rules themselves are usually specified using the notation of natural deduction. For example, the following typing rules specify the typing relation for a simple language of booleans: Each rule states that the conclusion below the line may be derived from the premises above the line. The first two rules have no premises above the line, so they are axioms. The third rule has premises above the line (specifically, three premises), so it is an inference rule. In programming languages, the type of a variable depends on where it is bound, which necessitates context-sensitive typing rules. These rules are given by a typing judgment, usually written , which states that an expression has type under a typing context that relates variables to their types. This notation can be used to give typing rules for variable references and lambda abstraction in the simply typed lambda calculus: Similarly, the following typing rule describes the construct of Standard ML: Not all systems of typing rules directly specify a type checking algorithm. For example, the typing rule for applying a parametrically polymorphic function in the Hindley–Milner type system requires "guessing" the appropriate type at which the function should be instantiated. Adapting a declarative rule system to a decidable algorithm requires the production of a separate, algorithmic system that can be proven to specify the same typing relation.

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Parametric polymorphism
In programming languages and type theory, parametric polymorphism allows a single piece of code to be given a "generic" type, using variables in place of actual types, and then instantiated with particular types as needed. Parametrically polymorphic functions and data types are sometimes called generic functions and generic datatypes, respectively, and they form the basis of generic programming. Parametric polymorphism may be contrasted with ad hoc polymorphism.
Subtyping
In programming language theory, subtyping (also subtype polymorphism or inclusion polymorphism) is a form of type polymorphism in which a subtype is a datatype that is related to another datatype (the supertype) by some notion of substitutability, meaning that program elements, typically subroutines or functions, written to operate on elements of the supertype can also operate on elements of the subtype. If S is a subtype of T, the subtyping relation (written as S
Type theory
In mathematics, logic, and computer science, a type theory is the formal presentation of a specific type system, and in general, type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a foundation of mathematics. Two influential type theories that were proposed as foundations are Alonzo Church's typed λ-calculus and Per Martin-Löf's intuitionistic type theory. Most computerized proof-writing systems use a type theory for their foundation, a common one is Thierry Coquand's Calculus of Inductive Constructions.

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