1781:, replaces the various fields with their subtypes. That is, the fields of the subtype are subtypes of the fields of the supertype. Since any operation supported for a field in the supertype is supported for its subtype, any operation feasible on the record supertype is supported by the record subtype. Depth subtyping only makes sense for immutable records: for example, you can assign 1.5 to the 'x' field of a real point (a record with two real fields), but you can't do the same to the 'x' field of an integer point (which, however, is a deep subtype of the real point type) because 1.5 is not an integer (see
565:, and an operator of comparison with an arbitrary Number is defined for both types, then values of either type can be passed to this function. However, the very possibility of implementing such an operator highly constrains the Number type (for example, one can't compare an integer with a complex number), and actually only comparing integers with integers, and reals with reals, makes sense. Rewriting this function so that it would only accept 'x' and 'y' of the same type requires
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742:. Predicates are partial functions that compare values to selection criteria. For example: "is an integer value greater than or equal to 100 and less than 200?". If a value matches the criteria then the function returns the value. If not, the value is not selected, and nothing is returned. (List comprehensions are a form of this pattern used in many programming languages.)
311:, which can also be stored in records, records types with subtyping provide some of the features of object-oriented programming. Typically, functional programming languages also provide some, usually restricted, form of parametric polymorphism. In a theoretical setting, it is desirable to study the interaction of the two features; a common theoretical setting is
3109:
Reynolds, John C. Using category theory to design implicit conversions and generic operators. In N. D. Jones, editor, Proceedings of the Aarhus
Workshop on Semantics-Directed Compiler Generation, number 94 in Lecture Notes in Computer Science. Springer-Verlag, January 1980. Also in Carl A. Gunter and
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from object-oriented languages; subtyping is a relation between types (interfaces in object-oriented parlance) whereas inheritance is a relation between implementations stemming from a language feature that allows new objects to be created from existing ones. In a number of object-oriented languages,
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A simple practical example of subtypes is shown in the diagram. The type "bird" has three subtypes "duck", "cuckoo" and "ostrich". Conceptually, each of these is a variety of the basic type "bird" that inherits many "bird" characteristics but has some specific differences. The
2936:
Cardelli, Luca. A semantics of multiple inheritance. In G. Kahn, D. MacQueen, and G. Plotkin, editors, Semantics of Data Types, volume 173 of
Lecture Notes in Computer Science, pages 51–67. Springer-Verlag, 1984. Full version in Information and Computation, 76(2/3):138–164,
1626:, in which the structure of two types determines whether or not one is a subtype of the other. The class-based object-oriented subtyping described above is nominal; a structural subtyping rule for an object-oriented language might say that if objects of type
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of the subtype (the number of situations where it can be accepted or introduced), as compared to its "more general" supertypes. The disadvantage of having this more detailed information is that it represents incorporated choices which reduce the
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Recall that a record is a collection of (named) fields. Since a subtype is a type which allows all operations allowed on the original type, a record subtype should support the same operations on the fields as the original type supported.
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extended with record types is perhaps the simplest theoretical setting in which a useful notion of subtyping may be defined and studied. Because the resulting calculus allows terms to have more than one type, it is no longer a "simple"
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The concept of subtyping has gained visibility (and synonymy with polymorphism in some circles) with the mainstream adoption of object-oriented programming. In this context, the principle of safe substitution is often called the
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In languages that allow side effects, like most object-oriented languages, subtyping is generally not sufficient to guarantee that a function can be safely used in the context of another. Liskov's work in this area focused on
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1695:. The subtyping induced by subclassing in an object-oriented language is usually inclusive; subtyping relations that relate integers and floating-point numbers, which are represented differently, are usually coercive.
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subtypes may be defined componentwise; in the case of width-extended records, type coercion simply discards any components which are not defined in the supertype. The type coercion for function types may be given by
1774:, adds more fields to the record. More formally, every (named) field appearing in the width supertype will appear in the width subtype. Thus, any operation feasible on the supertype will be supported by the subtype.
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In terms of information specificity, a subtype is considered more specific than any one of its supertypes, because it holds at least as much information as each of them. This may increase the applicability, or
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like records (structs, interfaces) or classes are defined intensionally by an explicit type declaration or by using an existing value, which encodes type information, as a prototype to be copied or extended.
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Subtyping and inheritance are compatible when all inherited fields and methods of the derived type have types which are subtypes of the corresponding fields and methods from the inherited type .
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by stating the membership of the set by a predicate over a domain of possible values. In common programming languages enumeration types are defined extensionally by listing values.
2038:. Informally, this reversal occurs because the refined type is "more liberal" in the types it accepts and "more conservative" in the type it returns. This is what exactly works in
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The conjunction of predicates has been expressed here through application of the second predicate over the domain of values conforming to the first predicate. Viewed as types,
2808:. Thus, when multiple subtyping relationships are defined, one must be careful to guarantee that all type coercions are coherent. For instance, if an integer such as 2 :
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Subtyping and inheritance are independent (orthogonal) relationships. They may coincide, but none is a special case of the other. In other words, between two types
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2989:, ACM Transactions on Programming Languages and Systems, Volume 16, Issue 6 (November 1994), pp. 1811–1841. An updated version appeared as CMU technical report:
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address at a conference on object-oriented programming in 1987. Because it must consider mutable objects, the ideal notion of subtyping defined by Liskov and
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notation is used in this diagram, with open-headed arrows showing the direction and type of the relationship between the supertype and its subtypes.
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is common in dynamically typed object-oriented languages. Sound structural subtyping rules for types other than object types are also well known.
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subtyping. These express two different ways of obtaining a new type of record that allows the same operations as the original record type.
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John C. Mitchell, editors, Theoretical
Aspects of Object-Oriented Programming: Types, Semantics, and Language Design (MIT Press, 1994).
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3047:, chapter 15 (subtyping of record types), 19.3 (nominal vs. structural types and subtyping), and 23.2 (varieties of polymorphism)
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In discussing the concept of subsumption, the set of values of a type is indicated by writing its name in mathematical italics:
318:. Various calculi that attempt to capture the theoretical properties of object-oriented programming may be derived from system F
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Due to the subtyping relation, a term may belong to more than one type. Subtyping is therefore a form of type polymorphism. In
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As a more practical example, a language might allow integer values to be used wherever floating point values are expected (
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1814:). The rule for width subtyping is reversed: every tag appearing in the width subtype must appear in the width supertype.
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where a term of type T is expected. The precise semantics of subtyping here crucially depends on the particulars of how
717:, then the two types are equal (although they may not be the same type if the type system distinguishes types by name).
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2005:{\displaystyle {T_{1}\leq :S_{1}\quad S_{2}\leq :T_{2}} \over {S_{1}\rightarrow S_{2}\leq :T_{1}\rightarrow T_{2}}}
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or functions), written to operate on elements of the supertype, can also operate on elements of the subtype.
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Copestake, Ann. Implementing typed feature structure grammars. Vol. 110. Stanford: CSLI publications, 2002.
2309:) are covariant, like return values. Mutable references which act as both sources and sinks are invariant.
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738:, which uses a predicate to define a set. Predicates can be defined over a domain (set of possible values)
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as an operand (parameter value or term) will therefore be able to operate over that value as one of type
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Example of subtypes: where bird is the supertype and all others are subtypes as denoted by the arrow in
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3077:. Proc. 17th ACM SIGPLAN-SIGACT Symp. on Principles of Programming Languages (POPL). pp. 125–135.
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is its return type; "−" before the type means the type is contravariant while "+" means covariant.
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1335:{\displaystyle {\mathit {Felinae=\{cat\in Felidae\mid \ ofSubfamily(cat,felinaeSubfamilyName)\}}}}
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The notion of subtyping in programming languages dates back to the 1960s; it was introduced in
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is similar to the treatment of parameter values and return values. Write-only references (or
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of a programming language essentially defines its own subtyping relation, which may well be
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Implementations of programming languages with subtyping fall into two general classes:
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In almost all type systems that define a subtyping relation, it is reflexive (meaning
1503:{\displaystyle {\mathit {Felis=\{cat\in Felinae\mid \ ofGenus(cat,felisGenusName)\}}}}
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2165:{\displaystyle {\mathtt {Function_{N}({-A_{1}},{-A_{2}},\dots ,{-A_{n}},{+B})}}}
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functions from subtype to supertype. For each subtyping relationship (
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2305:) are contravariant, like parameter values; read-only references (or
1102:{\displaystyle \mathbf {S} =\mathbf {T} \land P_{s}=P_{T}\land P_{s}}
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derivatives. The first formal treatments of subtyping were given by
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The coercion function is uniquely determined given the subtype and
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with subtyping of record types and is a theoretical basis for many
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than would be possible without it. Consider the following example:
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implementations, in which the representation of any value of type
1040:, so both must be true for a value to be selected. The predicate
956:{\displaystyle S=\{v\in D\mid \ P_{T}(v){\text{ and }}P_{s}(v)\}}
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The concept of subtyping is related to the linguistic notions of
307:. Since functional programming languages, by definition, support
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the term 'polymorphism' is commonly used to refer solely to this
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In coercive subtyping systems, subtypes are defined by implicit
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A type may be subsumed by more than one type: the supertypes of
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The second case can be illustrated by the relationship between
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2325:, all combinations of subtyping and inheritance are possible:
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is a function type, then a subtype of it is any function type
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3159:. Section 8.6 contrast the subtyping of records and objects.
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are the same. Since inheritance is an irreflexive relation,
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The first case is illustrated by independent types, such as
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can be coerced to a floating point number (say, 2.0 :
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is considerably stronger than what can be implemented in a
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Cook, William R.; Hill, Walter; Canning, Peter S. (1990).
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For example: there is a subfamily of cat species called
3000:"Behavioral Subtyping Using Invariants and Constraints"
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Component software: beyond object-oriented programming
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of parameter values and covariance of return values.
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2239:{\displaystyle {\mathtt {A_{1},A_{2},\dots ,A_{n}}}}
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can handle all of the messages that objects of type
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A type is a set of values. The set can be described
2863:would then be distinct from the identity coercion
2816:), then it is not admissible to coerce 2.1 :
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596:by listing all the values, or it can be described
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2385:. In most object oriented programming languages,
1634:can handle (that is, if they define all the same
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1770:One kind of way to achieve such support, called
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1806:Some systems also support subtyping of labeled
803:which applies additional criteria for the type
1594:to be applicable to values of all three types
807:, then sets for the two types can be defined:
772:which applies selection criteria for the type
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581:is used to define or evaluate whether a type
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2286:. This definition of subtyping is generally
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686:, the predicate which circumscribes the set
481:Programmers may take advantage of subtyping
2332:is neither a subtype nor a derived type of
1887:This can be summarised using the following
866:{\displaystyle T=\{v\in D\mid \ P_{T}(v)\}}
642:which it defines, is a superset of the set
194:, meaning that program elements (typically
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3172:
2350:is not a subtype but is a derived type of
2341:is a subtype but is not a derived type of
1683:implementations, in which a value of type
1614:Type theorists make a distinction between
235: ) means that any term of type S can
142:
128:
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2454:Learn how and when to remove this message
561:If integer and real are both subtypes of
184:that is related to another datatype (the
2494:, also note that the first parameter of
2359:is both a subtype and a derived type of
421:
2498:is this/self) and a derived class type
2468:function subtyping input contravariance
1788:Subtyping of records can be defined in
1667:also represents the same value at type
483:to write code in a more abstract manner
333:. It is also related to the concept of
14:
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2478:returning an object of the same type (
201:If S is a subtype of T, the subtyping
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2678:may be regarded as an object of type
2282:guaranteed by the supertypes in some
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694:(over the same domain) which defines
450:), or it might define a generic type
3129:, Cambridge University Press, 1998,
3054:, 2nd ed., Pearson Education, 2002,
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1609:
1138:belongs, is part of that subfamily.
1134:, to which the domestic cat species
3050:C. Szyperski, D. Gruntz, S. Murer,
2738:Coercion functions for records and
2466:The third case is a consequence of
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1710:) and transitive (meaning that if
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1028:as part of the compound predicate
994:{\displaystyle \mathbf {T} =P_{T}}
350:, with inheritance referred to as
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3127:Theories of programming languages
2538:must be a subtype of the type of
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1516:Felis <: Felinae <: Felidae
2986:A behavioral notion of subtyping
2983:Barbara Liskov, Jeannette Wing,
2824:, because the compound coercion
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2389:are unrelated by inheritance to
2290:, so it cannot be verified by a
2175:(which can be seen as a general
1801:functional programming languages
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690:, must be part of the predicate
478:are not subtypes of each other.
292:Functional programming languages
3734:Polymorphism (computer science)
3037:Types and programming languages
2972:Types and Programming Languages
2470:. Assume a super class of type
2397:can be considered a subtype of
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1650:from the other. This so-called
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2506:. By inheritance, the type of
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577:In type theory the concept of
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3252:Arbitrary-precision or bignum
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2950:Cook, Hill & Canning 1990
2882:Covariance and contravariance
2682:under the compound coercion (
2313:Relationship with inheritance
2263:{\displaystyle {\mathtt {B}}}
2246:are the parameter types, and
1755:give rise to the concepts of
1646:regardless of whether either
745:If there are two predicates,
620:means "is a subtype of", and
388:Liskov substitution principle
294:often allow the subtyping of
3075:Inheritance is not subtyping
2629:is provided, and any object
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2578:, which is only possible if
1803:that support both features.
300:simply typed lambda calculus
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3744:Object-oriented programming
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2434:the claims made and adding
624:means "is a supertype of".
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337:in mathematical logic (see
272:object-oriented programming
156:programming language theory
101:Single and dynamic dispatch
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2637:is regarded as the object
1777:The second method, called
646:, so that every member of
616:. The conventional symbol
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352:implementation inheritance
278:, while the techniques of
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3593:Strongly typed identifier
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1747:Width and depth subtyping
1036:. The two predicates are
27:Form of type polymorphism
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2183:-like languages), where
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394:who popularized it in a
3668:Parametric polymorphism
2894:Class-based programming
2617:), a coercion function
1854:with the property that
1797:parametric polymorphism
1689:automatically converted
1109:subsumes the predicate
280:parametric polymorphism
251:are defined by a given
64:Parametric polymorphism
2888:circle-ellipse problem
2712:from a type to itself
2590:can't be a subtype of
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2014:The parameter type of
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1583:{\displaystyle s\in T}
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1553:{\displaystyle s\in S}
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335:bounded quantification
168:inclusion polymorphism
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1021:{\displaystyle P_{s}}
1001:is applied alongside
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796:{\displaystyle P_{s}}
767:
765:{\displaystyle P_{T}}
638:if the set of values
585:is a subtype of type
425:
412:§ Function types
348:interface inheritance
3035:Benjamin C. Pierce,
2970:Benjamin C. Pierce,
2961:Pierce, ch. 15 notes
2909:Behavioral subtyping
2797:))), reflecting the
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1623:structural subtyping
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736:Set-builder notation
650:is also a member of
567:bounded polymorphism
414:below for details.)
404:behavioral subtyping
376:implicit conversions
346:subtyping is called
282:would be considered
276:subtype polymorphism
257:programming language
190:) by some notion of
164:subtype polymorphism
55:Operator overloading
50:Function overloading
3673:Primitive data type
3578:Recursive data type
3431:Algebraic data type
3307:Quadruple precision
3149:A theory of objects
3093:10.1145/96709.96721
3039:, MIT Press, 2002,
2526:to be a subtype of
1734:). This makes it a
285:generic programming
78:Generic programming
41:Ad hoc polymorphism
3636:Abstract data type
3317:Extended precision
3276:Reduced precision
3151:, Springer, 1996,
2546:, in other words:
2419:possibly contains
2299:mutable references
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602:User-defined types
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339:Order-sorted logic
116:Predicate dispatch
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3448:Associative array
3312:Octuple precision
2727:identity function
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2463:
2456:
2421:original research
2297:The subtyping of
2002:
1795:, which combines
1730: <:
1722: <:
1714: <:
1702: <:
1691:into one of type
1675: <:
1617:nominal subtyping
1610:Subtyping schemes
1413:
1215:
929:
906:
840:
370:in 1980 who used
309:function literals
172:type polymorphism
152:
151:
111:Multiple dispatch
16:(Redirected from
3751:
3688:Type constructor
3573:Opaque data type
3505:Record or Struct
3302:Double precision
3297:Single precision
3188:
3181:
3174:
3165:
3164:
3123:John C. Reynolds
3106:
3086:
3017:
3016:
3014:
3013:
3008:
2981:
2975:
2968:
2962:
2959:
2953:
2947:
2938:
2934:
2928:
2925:
2522:. In order for
2474:having a method
2459:
2452:
2448:
2445:
2439:
2436:inline citations
2412:
2411:
2404:
2400:
2396:
2392:
2388:
2384:
2380:
2373:
2369:
2269:
2267:
2266:
2261:
2259:
2258:
2245:
2243:
2242:
2237:
2235:
2234:
2233:
2232:
2214:
2213:
2201:
2200:
2171:
2169:
2168:
2163:
2161:
2160:
2156:
2145:
2144:
2143:
2121:
2120:
2119:
2103:
2102:
2101:
2085:
2084:
2029:
2011:
2009:
2008:
2003:
2001:
2000:
1999:
1987:
1986:
1974:
1973:
1961:
1960:
1949:
1948:
1947:
1935:
1934:
1924:
1923:
1911:
1910:
1899:
1886:
1869:
1853:
1837:
1642:is a subtype of
1589:
1587:
1586:
1581:
1559:
1557:
1556:
1551:
1518:
1509:
1507:
1506:
1501:
1499:
1498:
1341:
1339:
1338:
1333:
1331:
1330:
1119:
1108:
1106:
1105:
1100:
1098:
1097:
1085:
1084:
1072:
1071:
1059:
1051:
1035:
1027:
1025:
1024:
1019:
1017:
1016:
1000:
998:
997:
992:
990:
989:
977:
962:
960:
959:
954:
940:
939:
930:
927:
916:
915:
904:
872:
870:
869:
864:
850:
849:
838:
802:
800:
799:
794:
792:
791:
771:
769:
768:
763:
761:
760:
741:
697:
689:
681:
664:
653:
649:
645:
641:
611:
564:
557:
554:
551:
548:
545:
542:
539:
536:
533:
530:
527:
524:
521:
518:
515:
512:
509:
506:
503:
500:
497:
494:
491:
477:
473:
469:
465:
461:
457:
453:
449:
445:
368:John C. Reynolds
298:. Consequently,
245:"safely be used"
234:
224:
214:
192:substitutability
144:
137:
130:
96:Virtual function
73:Generic function
30:
29:
21:
3759:
3758:
3754:
3753:
3752:
3750:
3749:
3748:
3719:
3718:
3717:
3712:
3693:Type conversion
3628:
3622:
3558:Enumerated type
3531:
3417:
3411:null-terminated
3387:
3352:
3240:
3197:
3192:
3162:
3118:
3116:Further reading
3113:
3103:
3084:10.1.1.102.8635
3065:
3026:
3021:
3020:
3011:
3009:
3002:
2996:Wing, Jeannette
2992:Liskov, Barbara
2982:
2978:
2969:
2965:
2960:
2956:
2948:
2941:
2935:
2931:
2926:
2922:
2917:
2904:Refinement type
2878:
2871:
2862:
2849:
2836:
2792:
2791:
2784:
2769:
2768:
2761:
2734:
2724:
2707:
2694:
2649:
2607:type conversion
2603:
2460:
2449:
2443:
2440:
2425:
2413:
2409:
2398:
2394:
2390:
2386:
2382:
2378:
2371:
2367:
2315:
2254:
2253:
2251:
2248:
2247:
2228:
2224:
2209:
2205:
2196:
2192:
2191:
2190:
2188:
2185:
2184:
2149:
2139:
2135:
2131:
2115:
2111:
2107:
2097:
2093:
2089:
2080:
2076:
2054:
2053:
2051:
2048:
2047:
2028:
2021:
2015:
1995:
1991:
1982:
1978:
1969:
1965:
1956:
1952:
1951:
1943:
1939:
1930:
1926:
1919:
1915:
1906:
1902:
1901:
1898:
1895:
1894:
1884:
1877:
1871:
1868:
1861:
1855:
1852:
1845:
1839:
1836:
1829:
1823:
1820:
1810:types (such as
1793:
1779:depth subtyping
1772:width subtyping
1749:
1744:
1612:
1569:
1566:
1565:
1539:
1536:
1535:
1514:
1353:
1352:
1350:
1347:
1346:
1149:
1148:
1146:
1143:
1142:
1114:
1093:
1089:
1080:
1076:
1067:
1063:
1055:
1047:
1045:
1042:
1041:
1033:
1012:
1008:
1006:
1003:
1002:
985:
981:
973:
971:
968:
967:
935:
931:
928: and
926:
911:
907:
881:
878:
877:
845:
841:
815:
812:
811:
787:
783:
781:
778:
777:
756:
752:
750:
747:
746:
739:
695:
687:
673:
672:(and therefore
662:
651:
647:
643:
639:
609:
575:
562:
559:
558:
555:
552:
549:
546:
543:
540:
537:
534:
531:
528:
525:
522:
519:
516:
513:
510:
507:
504:
501:
498:
495:
492:
489:
475:
471:
467:
463:
459:
455:
451:
447:
443:
420:
372:category theory
360:
321:
316:
226:
216:
206:
170:) is a form of
148:
106:Double dispatch
28:
23:
22:
15:
12:
11:
5:
3757:
3747:
3746:
3741:
3736:
3731:
3714:
3713:
3711:
3710:
3705:
3700:
3695:
3690:
3685:
3680:
3675:
3670:
3665:
3664:
3663:
3653:
3648:
3646:Data structure
3643:
3638:
3632:
3630:
3624:
3623:
3621:
3620:
3615:
3610:
3605:
3600:
3595:
3590:
3585:
3580:
3575:
3570:
3565:
3560:
3555:
3550:
3545:
3539:
3537:
3533:
3532:
3530:
3529:
3528:
3527:
3517:
3512:
3507:
3502:
3497:
3492:
3491:
3490:
3480:
3475:
3470:
3465:
3460:
3455:
3450:
3445:
3440:
3439:
3438:
3427:
3425:
3419:
3418:
3416:
3415:
3414:
3413:
3403:
3397:
3395:
3389:
3388:
3386:
3385:
3380:
3379:
3378:
3373:
3362:
3360:
3354:
3353:
3351:
3350:
3345:
3340:
3339:
3338:
3328:
3327:
3326:
3325:
3324:
3314:
3309:
3304:
3299:
3294:
3293:
3292:
3287:
3285:Half precision
3282:
3272:Floating point
3269:
3264:
3259:
3254:
3248:
3246:
3242:
3241:
3239:
3238:
3233:
3228:
3223:
3218:
3213:
3207:
3205:
3199:
3198:
3191:
3190:
3183:
3176:
3168:
3161:
3160:
3138:
3119:
3117:
3114:
3112:
3111:
3101:
3071:
3064:
3063:
3048:
3032:
3025:
3022:
3019:
3018:
2976:
2963:
2954:
2939:
2929:
2919:
2918:
2916:
2913:
2912:
2911:
2906:
2901:
2896:
2891:
2884:
2877:
2874:
2867:
2854:
2841:
2828:
2799:contravariance
2789:
2782:
2778:
2766:
2759:
2755:
2740:disjoint union
2732:
2716:
2699:
2686:
2641:
2602:
2599:
2462:
2461:
2416:
2414:
2407:
2364:
2363:
2354:
2345:
2336:
2314:
2311:
2257:
2231:
2227:
2223:
2220:
2217:
2212:
2208:
2204:
2199:
2195:
2159:
2155:
2152:
2148:
2142:
2138:
2134:
2130:
2127:
2124:
2118:
2114:
2110:
2106:
2100:
2096:
2092:
2088:
2083:
2079:
2075:
2072:
2069:
2066:
2063:
2060:
2057:
2030:is said to be
2026:
2019:
1998:
1994:
1990:
1985:
1981:
1977:
1972:
1968:
1964:
1959:
1955:
1946:
1942:
1938:
1933:
1929:
1922:
1918:
1914:
1909:
1905:
1882:
1875:
1866:
1859:
1850:
1843:
1834:
1827:
1819:
1818:Function types
1816:
1808:disjoint union
1791:
1748:
1745:
1743:
1740:
1611:
1608:
1579:
1576:
1573:
1549:
1546:
1543:
1532:T :> S
1511:
1510:
1497:
1494:
1491:
1488:
1485:
1482:
1479:
1476:
1473:
1470:
1467:
1464:
1461:
1458:
1455:
1452:
1449:
1446:
1443:
1440:
1437:
1434:
1431:
1428:
1425:
1422:
1419:
1416:
1410:
1407:
1404:
1401:
1398:
1395:
1392:
1389:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1365:
1362:
1359:
1356:
1343:
1342:
1329:
1326:
1323:
1320:
1317:
1314:
1311:
1308:
1305:
1302:
1299:
1296:
1293:
1290:
1287:
1284:
1281:
1278:
1275:
1272:
1269:
1266:
1263:
1260:
1257:
1254:
1251:
1248:
1245:
1242:
1239:
1236:
1233:
1230:
1227:
1224:
1221:
1218:
1212:
1209:
1206:
1203:
1200:
1197:
1194:
1191:
1188:
1185:
1182:
1179:
1176:
1173:
1170:
1167:
1164:
1161:
1158:
1155:
1152:
1096:
1092:
1088:
1083:
1079:
1075:
1070:
1066:
1062:
1058:
1054:
1050:
1015:
1011:
988:
984:
980:
976:
966:The predicate
964:
963:
952:
949:
946:
943:
938:
934:
925:
922:
919:
914:
910:
903:
900:
897:
894:
891:
888:
885:
874:
873:
862:
859:
856:
853:
848:
844:
837:
834:
831:
828:
825:
822:
819:
790:
786:
759:
755:
719:
718:
699:
666:
655:
574:
571:
488:
419:
416:
400:Jeannette Wing
392:Barbara Liskov
359:
356:
319:
314:
253:type formalism
237:safely be used
150:
149:
147:
146:
139:
132:
124:
121:
120:
119:
118:
113:
108:
103:
98:
90:
89:
83:
82:
81:
80:
75:
67:
66:
60:
59:
58:
57:
52:
44:
43:
37:
36:
26:
9:
6:
4:
3:
2:
3756:
3745:
3742:
3740:
3737:
3735:
3732:
3730:
3727:
3726:
3724:
3709:
3706:
3704:
3701:
3699:
3696:
3694:
3691:
3689:
3686:
3684:
3681:
3679:
3676:
3674:
3671:
3669:
3666:
3662:
3659:
3658:
3657:
3654:
3652:
3649:
3647:
3644:
3642:
3639:
3637:
3634:
3633:
3631:
3625:
3619:
3616:
3614:
3611:
3609:
3606:
3604:
3601:
3599:
3596:
3594:
3591:
3589:
3586:
3584:
3581:
3579:
3576:
3574:
3571:
3569:
3568:Function type
3566:
3564:
3561:
3559:
3556:
3554:
3551:
3549:
3546:
3544:
3541:
3540:
3538:
3534:
3526:
3523:
3522:
3521:
3518:
3516:
3513:
3511:
3508:
3506:
3503:
3501:
3498:
3496:
3493:
3489:
3486:
3485:
3484:
3481:
3479:
3476:
3474:
3471:
3469:
3466:
3464:
3461:
3459:
3456:
3454:
3451:
3449:
3446:
3444:
3441:
3437:
3434:
3433:
3432:
3429:
3428:
3426:
3424:
3420:
3412:
3409:
3408:
3407:
3404:
3402:
3399:
3398:
3396:
3394:
3390:
3384:
3381:
3377:
3374:
3372:
3369:
3368:
3367:
3364:
3363:
3361:
3359:
3355:
3349:
3346:
3344:
3341:
3337:
3334:
3333:
3332:
3329:
3323:
3320:
3319:
3318:
3315:
3313:
3310:
3308:
3305:
3303:
3300:
3298:
3295:
3291:
3288:
3286:
3283:
3281:
3278:
3277:
3275:
3274:
3273:
3270:
3268:
3265:
3263:
3260:
3258:
3255:
3253:
3250:
3249:
3247:
3243:
3237:
3234:
3232:
3229:
3227:
3224:
3222:
3219:
3217:
3214:
3212:
3209:
3208:
3206:
3204:
3203:Uninterpreted
3200:
3196:
3189:
3184:
3182:
3177:
3175:
3170:
3169:
3166:
3158:
3157:0-387-94775-2
3154:
3150:
3146:
3145:Luca Cardelli
3142:
3139:
3137:, chapter 16.
3136:
3135:0-521-59414-6
3132:
3128:
3124:
3121:
3120:
3108:
3107:
3104:
3102:0-89791-343-4
3098:
3094:
3090:
3085:
3080:
3076:
3070:
3069:
3061:
3060:0-201-74572-0
3057:
3053:
3049:
3046:
3045:0-262-16209-1
3042:
3038:
3034:
3033:
3031:
3030:
3006:
3001:
2998:(July 1999).
2997:
2993:
2988:
2987:
2980:
2973:
2967:
2958:
2951:
2946:
2944:
2933:
2924:
2920:
2910:
2907:
2905:
2902:
2900:
2897:
2895:
2892:
2889:
2885:
2883:
2880:
2879:
2873:
2870:
2866:
2861:
2857:
2853:
2848:
2844:
2840:
2835:
2831:
2827:
2823:
2819:
2815:
2811:
2807:
2802:
2800:
2796:
2788:
2781:
2777:
2773:
2765:
2758:
2754:
2750:
2746:
2741:
2736:
2731:
2728:
2723:
2719:
2715:
2711:
2710:type coercion
2706:
2702:
2698:
2693:
2689:
2685:
2681:
2677:
2673:
2669:
2665:
2661:
2657:
2653:
2648:
2644:
2640:
2636:
2632:
2628:
2624:
2620:
2616:
2612:
2608:
2598:
2595:
2593:
2589:
2585:
2581:
2577:
2573:
2569:
2565:
2561:
2557:
2553:
2549:
2545:
2541:
2537:
2533:
2529:
2525:
2521:
2517:
2513:
2509:
2505:
2501:
2497:
2493:
2489:
2485:
2481:
2477:
2473:
2469:
2458:
2455:
2447:
2437:
2433:
2429:
2423:
2422:
2417:This article
2415:
2406:
2405:
2402:
2375:
2362:
2358:
2355:
2353:
2349:
2346:
2344:
2340:
2337:
2335:
2331:
2328:
2327:
2326:
2324:
2320:
2310:
2308:
2304:
2300:
2295:
2293:
2289:
2285:
2281:
2277:
2271:
2218:
2182:
2178:
2174:
2125:
2045:
2041:
2037:
2033:
2032:contravariant
2025:
2018:
2012:
1996:
1992:
1983:
1979:
1975:
1970:
1966:
1957:
1953:
1944:
1940:
1936:
1931:
1927:
1920:
1916:
1912:
1907:
1903:
1892:
1890:
1881:
1874:
1865:
1858:
1849:
1842:
1833:
1826:
1815:
1813:
1809:
1804:
1802:
1798:
1794:
1786:
1784:
1780:
1775:
1773:
1768:
1764:
1762:
1758:
1754:
1739:
1737:
1733:
1729:
1725:
1721:
1717:
1713:
1709:
1706:for any type
1705:
1701:
1696:
1694:
1690:
1686:
1682:
1678:
1674:
1670:
1666:
1662:
1657:
1655:
1654:
1649:
1645:
1641:
1637:
1633:
1629:
1625:
1624:
1619:
1618:
1607:
1605:
1601:
1597:
1593:
1577:
1574:
1571:
1563:
1547:
1544:
1541:
1533:
1529:
1525:
1520:
1517:
1408:
1384:
1345:
1344:
1210:
1186:
1141:
1140:
1139:
1137:
1133:
1129:
1125:
1120:
1117:
1112:
1094:
1090:
1086:
1081:
1077:
1073:
1068:
1064:
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225:, or
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