Editing Subtyping

{{short description|Form of type polymorphism}}
{{Polymorphism}}

In [[programming language theory]], '''subtyping''' (also called '''subtype polymorphism''' or '''inclusion polymorphism''') is a form of [[Polymorphism (computer science)|type polymorphism]]. 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 [[binary relation|relation]] (written as {{math|1=''S'' &lt;: ''T''}}, &nbsp;{{math|1=''S'' ⊑ ''T''}},<ref>Copestake, Ann. Implementing typed feature structure grammars. Vol. 110. Stanford: CSLI publications, 2002.</ref> or &nbsp;{{math|1=''S'' ≤: ''T''}}&nbsp;) means that any term of type S can ''safely be used'' in ''any context'' where a term of type T is expected. The precise semantics of subtyping here crucially depends on the particulars of how ''"safely be used"'' and ''"any context"'' are defined by a given [[Formal_language|type formalism]] or [[programming language]]. The [[type system]] of a programming language essentially defines its own subtyping relation, which may well be [[identity relation|trivial]], should the language support no (or very little) conversion mechanisms.

Due to the subtyping relation, a term may belong to more than one type. Subtyping is therefore a form of type polymorphism. In [[object-oriented programming]] the term 'polymorphism' is commonly used to refer solely to this ''subtype polymorphism'', while the techniques of [[parametric polymorphism]] would be considered ''[[generic programming]]''.

[[Functional programming languages]] often allow the subtyping of [[Record (computer science)|records]]. Consequently, [[simply typed lambda calculus]] extended with record types is perhaps the simplest theoretical setting in which a useful notion of subtyping may be defined and studied.<ref>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, 1988.</ref> Because the resulting calculus allows terms to have more than one type, it is no longer a "simple" [[type theory]]. Since functional programming languages, by definition, support [[function literals]], 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 [[system F-sub|system F<sub>&lt;:</sub>]]. Various calculi that attempt to capture the theoretical properties of object-oriented programming may be derived from system F<sub>&lt;:</sub>.

The concept of subtyping is related to the linguistic notions of [[hyponymy]] and [[holonymy]]. It is also related to the concept of [[bounded quantification]] in mathematical logic (see [[Order-sorted logic]]). Subtyping should not be confused with the notion of (class or object) [[inheritance (object-oriented programming)|inheritance]] from object-oriented languages;{{sfn|Cook|Hill|Canning|1990}} 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, subtyping is called '''interface inheritance''',<!-- todo add the insightful example from Mitchell with stack/queue/dequeue in the body --> with inheritance referred to as ''implementation inheritance''.

== Origins ==
The notion of subtyping in programming languages dates back to the 1960s; it was introduced in [[Simula]] derivatives. The first formal treatments of subtyping were given by [[John C. Reynolds]] in 1980 who used [[category theory]] to formalize [[implicit conversion]]s, and [[Luca Cardelli]] (1985).<ref>Pierce, ch. 15 notes</ref>

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 [[Liskov substitution principle]], after [[Barbara Liskov]] who popularized it in a [[keynote]] 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 [[Jeannette Wing]], called [[behavioral subtyping]] is considerably stronger than what can be implemented in a [[type checker]]. (See {{section link||Function types}} below for details.)

== Examples ==
[[Image:Inheritance.svg|thumb|right|350px|Example of subtypes: where bird is the supertype and all others are subtypes as denoted by the arrow in [[Unified Modeling Language|UML]] notation]]
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 [[Unified Modeling Language|UML]] notation is used in this diagram, with open-headed arrows showing the direction and type of the relationship between the supertype and its subtypes.

As a more practical example, a language might allow integer values to be used wherever floating point values are expected (<code>Integer</code> <: <code>Float</code>), or it might define a generic type <samp>Number</samp> as a common supertype of integers and the reals. In this second case, we only have <code>Integer</code> <: <code>Number</code> and <code>Float</code> <: <code>Number</code>, but <code>Integer</code> and <code>Float</code> are not subtypes of each other.

Programmers may take advantage of subtyping [[abstraction principle (programming)|to write code in a more abstract manner]] than would be possible without it. Consider the following example:

<syntaxhighlight lang="vbnet">
function max (x as Number, y as Number) is
    if x < y then
        return y
    else
        return x
end
</syntaxhighlight>

If integer and real are both subtypes of <code>Number</code>, 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 [[bounded polymorphism]].

Subtyping enables a given type to be substituted for another type or abstraction. Subtyping is said to establish an [[is-a]] relationship between the subtype and some existing abstraction, either implicitly or explicitly, depending on language support. The relationship can be expressed explicitly via inheritance in languages that support inheritance as a subtyping mechanism.

===C++===
The following C++ code establishes an explicit inheritance relationship between classes '''B''' and '''A''', where '''B''' is both a subclass and a subtype of '''A''', and can be used as an '''A''' wherever a '''B''' is specified (via a reference, a pointer or the object itself).

<syntaxhighlight lang=cpp>class A
{ public:
   void DoSomethingALike() const {}
};

class B : public A
{ public:
   void DoSomethingBLike() const {}
};

void UseAnA(A const& some_A)
{
   some_A.DoSomethingALike();
}

void SomeFunc()
{
   B b;
   UseAnA(b); // b can be substituted for an A.
}
</syntaxhighlight><ref name="Mitchell2002">
{{cite book
 | last=Mitchell
 | first=John
 | author-link=John C. Mitchell
 | title=Concepts in programming language
 | year=2002
 | publisher=Cambridge University Press
 | location=Cambridge, UK
 | isbn=0-521-78098-5
 | page=287
 | chapter=10 "Concepts in object-oriented languages"}}
</ref>

===Python===
The following python code establishes an explicit inheritance relationship between classes {{var|B}} and {{var|A}}, where {{var|B}} is both a subclass and a subtype of {{var|A}}, and can be used as an {{var|A}} wherever a {{var|B}} is required.

<syntaxhighlight lang="python">
class A:
    def do_something_a_like(self):
        pass

class B(A):
    def do_something_b_like(self):
        pass

def use_an_a(some_a):
    some_a.do_something_a_like()

def some_func():
    b = B()
    use_an_a(b)  # b can be substituted for an A.
</syntaxhighlight>

The following example, {{var|type(a)}} is a "regular" type, and {{var|type(type(a))}} is a metatype. While as distributed all types have the same metatype ({{var|PyType_Type}}, which is also its own metatype), this is not a requirement. The type of classic classes, known as {{var|types.ClassType}}, can also be considered a distinct metatype.<ref>{{cite web|author=Guido van Rossum|title=Subtyping Built-in Types|url=https://www.python.org/dev/peps/pep-0253/|access-date=2 October 2012}}</ref>

<syntaxhighlight lang="pycon">
>>> a = 0
>>> type(a)
<type 'int'>
>>> type(type(a))
<type 'type'>
>>> type(type(type(a)))
<type 'type'>
>>> type(type(type(type(a))))
<type 'type'>
</syntaxhighlight>

===Java===

In Java, '''is-a''' relation between the type parameters of one class or interface and the type parameters of another are determined by the extends and [[Interface (Java)|implements]] clauses.

Using the {{code|Collections}} classes, {{code|ArrayList<E>}} implements {{code|List<E>}}, and {{code|List<E>}} extends {{code|Collection<E>}}. So {{code|ArrayList<String>}} is a subtype of {{code|List<String>}}, which is a subtype of {{code|Collection<String>}}. The subtyping relationship is preserved between the types automatically. When defining an interface, {{code|PayloadList}}, that associates an optional value of [[generic type]] P with each element, its declaration might look like:

<syntaxhighlight lang=java>
interface PayloadList<E, P> extends List<E> {
    void setPayload(int index, P val);
    ...
}
</syntaxhighlight>

The following parameterizations of PayloadList are subtypes of {{code|List<String>}}:

<syntaxhighlight lang=java>
PayloadList<String, String>
PayloadList<String, Integer>
PayloadList<String, Exception>
</syntaxhighlight>

== Subsumption ==

In type theory the concept of ''subsumption''<ref>Benjamin C. Pierce, ''Types and Programming Languages'', MIT Press, 2002, 15.1 "Subsumption", p. 181-182</ref>  is used to define or evaluate whether a type '''S''' is a subtype of type '''T'''.

A type is a set of values. The set can be described ''extensionally'' by listing all the values, or it can be described ''intensionally'' 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. [[User-defined type]]s 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.

In discussing the concept of subsumption, the set of values of a type is indicated by writing its name in mathematical italics: {{mvar|T}}. The type, viewed as a predicate over a domain, is indicated by writing its name in bold: '''T'''. The conventional symbol '''<:''' means "is a subtype of", and ''':>''' means "is a supertype of".{{Citation needed|reason=Are &lt;: and &gt;: from the Julia language? Older usage? I don't see these symbols in Unicode or Latex|date=March 2024}}

* A type '''T''' ''subsumes'' '''S''' if the set of values {{mvar|T}} which it defines, is a superset of the set {{mvar|S}}, so that every member of {{mvar|S}} is also a member of {{mvar|T}}. 
* A type may be subsumed by more than one type: the supertypes of '''S''' intersect at {{mvar|S}}.
* If '''S <: T''' (and therefore {{math|''S'' ⊆ ''T'' }}), then '''T''', the predicate which circumscribes the set {{mvar|T}}, must be part of the predicate '''S''' (over the same domain) which defines {{mvar|S}}. 
* If '''S''' subsumes '''T''', and '''T''' subsumes '''S''', then the two types are equal (although they may not be the same type if the type system distinguishes types by name).

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 ''relevance'' 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 ''prevalence'' of the subtype (the number of situations which are able to generate or produce it).

In the context of subsumption, the type definitions can be expressed using [[Set-builder notation]], which uses a predicate to define a set. Predicates can be defined over a domain (set of possible values) {{mvar|D}}. 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.)

If there are two predicates, <math>P_T</math> which applies selection criteria for the type '''T''', and <math>P_s</math> which applies additional criteria for the type '''S''', then sets for the two types can be defined:

:<math>T = \{v \in D  \mid \ P_T(v)\}</math>

:<math>S =\{v \in D  \mid \ P_T(v)\text{ and }P_s(v)\}</math>
  
The predicate <math>\mathbf T = P_T</math> is applied alongside <math>P_s</math> as part of the compound predicate '''S''' defining {{mvar|S}}. The two predicates are ''conjoined'', so both must be true for a value to be selected. The predicate <math>\mathbf S = \mathbf T \land P_s = P_T \land P_s</math> subsumes the predicate '''T''', so {{nowrap|'''S <: T'''.}}

For example: there is a subfamily of cat species called ''Felinae'', which is part of the family ''Felidae''. The genus ''Felis'', to which the domestic cat species ''Felis catus'' belongs, is part of that subfamily.

:<math>\mathit{Felinae = \{cat \in Felidae  \mid \ ofSubfamily(cat, felinaeSubfamilyName)\}}</math>

:<math>\mathit{Felis =\{cat \in Felinae \mid \ ofGenus(cat, felisGenusName)\}}</math>

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, {{nowrap|'''Felis <: Felinae <: Felidae'''}}.

If '''T''' subsumes '''S''' ('''T :> S''') then a procedure, function or expression given a value <math>s \in S</math> as an operand (parameter value or term) will therefore be able to operate over that value as one of type '''T''', because <math>s \in T</math>. In the example above, we could expect the function ''ofSubfamily'' to be applicable to values of all three types '''Felidae''', '''Felinae''' and '''Felis'''.

== Subtyping schemes ==

Type theorists make a distinction between '''[[nominal type system|nominal subtyping]]''', in which only types declared in a certain way may be subtypes of each other, and '''[[structural type system|structural subtyping]]''', 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 ''A'' can handle all of the messages that objects of type ''B'' can handle (that is, if they define all the same [[Method (computer science)|method]]s), then ''A'' is a subtype of ''B'' regardless of whether either [[inheritance (object-oriented programming)|inherits]] from the other. This so-called ''[[duck typing]]'' is common in dynamically typed object-oriented languages. Sound structural subtyping rules for types other than object types are also well known.{{citation needed|date=June 2012}}

Implementations of programming languages with subtyping fall into two general classes: ''inclusive'' implementations, in which the representation of any value of type ''A'' also represents the same value at type ''B'' if ''A''&nbsp;&lt;:&nbsp;''B'', and ''coercive'' implementations, in which a value of type ''A'' can be ''automatically converted'' into one of type ''B''. 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.

In almost all type systems that define a subtyping relation, it is reflexive (meaning ''A''&nbsp;&lt;:&nbsp;''A'' for any type ''A'') and transitive (meaning that if ''A''&nbsp;&lt;:&nbsp;''B'' and ''B''&nbsp;&lt;:&nbsp;''C'' then ''A''&nbsp;&lt;:&nbsp;''C''). This makes it a [[preorder]] on types.

== Record types ==

=== Width and depth subtyping ===

Types of [[Record (computer science)|records]] give rise to the concepts of ''width'' and ''depth'' subtyping. These express two different ways of obtaining a new type of record that allows the same operations as the original record type.

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.

One kind of way to achieve such support, called ''width subtyping'', 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.

The second method, called ''depth subtyping'', 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 [[Covariance and contravariance (computer science)|Variance]]).

Subtyping of records can be defined in [[System F-sub|System F<sub><:</sub>]], which combines [[parametric polymorphism]] with subtyping of record types and is a theoretical basis for many [[functional programming languages]] that support both features.

Some systems also support subtyping of labeled [[disjoint union]] types (such as [[algebraic data type]]s). The rule for width subtyping is reversed: every tag appearing in the width subtype must appear in the width supertype.

==Function types==

If {{math|1=''T''<sub>1</sub> → ''T''<sub>2</sub>}} is a function type, then a subtype of it is any function type {{math|1=''S''<sub>1</sub> → ''S''<sub>2</sub>}} with the property that {{math|1=''T''<sub>1</sub> <: ''S''<sub>1</sub>}} and {{math|1=''S''<sub>2</sub> <: ''T''<sub>2</sub>.}} This can be summarised using the following [[typing rule]]:

<math display="block">{T_1 \leq: S_1 \quad S_2 \leq: T_2}
\over
{S_1 \rightarrow S_2 \leq: T_1 \rightarrow T_2}
</math>

The parameter type of {{math|1=''S''<sub>1</sub> → ''S''<sub>2</sub>}} is said to be [[Covariance and contravariance (computer science)|contravariant]] because the subtyping relation is reversed for it, whereas the return type is [[Covariance and contravariance (computer science)|covariant]]. 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 [[Scala (programming language)|Scala]]: a ''n''-ary function is internally a class that inherits the <math>\mathtt{ Function_N({-A_1}, {-A_2}, \dots, {-A_n}, {+B})}</math> [[Trait (computer programming)|trait]] (which can be seen as a general [[Application programming interface|interface]] in [[Java (programming language)|Java]]-like languages), where <math>\mathtt{A_1, A_2, \dots, A_n}</math> are the parameter types, and <math>\mathtt{B}</math> is its return type; "−" before the type means the type is contravariant while "+" means covariant.

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 [[behavioral subtyping]], which besides the type system safety discussed in this article also requires that subtypes preserve all [[Invariant (computer science)|invariants]] guaranteed by the supertypes in some [[Design by Contract|contract]].<ref name="LSP">Barbara Liskov, Jeannette Wing, ''[http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.39.1223 A behavioral notion of subtyping]'', ACM Transactions on Programming Languages and Systems, Volume 16, Issue 6 (November 1994), pp. 1811–1841. An updated version appeared as CMU technical report: {{cite web|url=http://reports-archive.adm.cs.cmu.edu/anon/1999/CMU-CS-99-156.ps|title=Behavioral Subtyping Using Invariants and Constraints|last=Liskov|first=Barbara|author-link=Barbara Liskov|author2=Wing, Jeannette |author-link2=Jeannette Wing |date=July 1999|format=[[PostScript|PS]]|access-date=2006-10-05}}</ref> This definition of subtyping is generally [[undecidable problem|undecidable]], so it cannot be verified by a [[type checker]].

The subtyping of [[Immutable object|mutable reference]]s is similar to the treatment of parameter values and return values. Write-only references (or ''sinks'') are contravariant, like parameter values; read-only references (or ''sources'') are covariant, like return values. Mutable references which act as both sources and sinks are invariant.

==Relationship with inheritance==

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 ''S'' and ''T'', all combinations of subtyping and inheritance are possible:
# ''S'' is neither a subtype nor a derived type of ''T''
# ''S'' is a subtype but is not a derived type of ''T''
# ''S'' is not a subtype but is a derived type of ''T''
# ''S'' is both a subtype and a derived type of ''T''
The first case is illustrated by independent types, such as <code>Boolean</code> and <code>Float</code>.

The second case can be illustrated by the relationship between <code>Int32</code> and <code>Int64</code>. In most object oriented programming languages, <code>Int64</code> are unrelated by inheritance to <code>Int32</code>. However <code>Int32</code> can be considered a subtype of <code>Int64</code> since any 32 bit integer value can be promoted into a 64 bit integer value.

{{Original research|date=July 2022}}
The third case is a consequence of [[Subtyping of functions|function subtyping input contravariance]]. Assume a super class of type ''T'' having a method ''m'' returning an object of the same type (''i.e.'' the type of ''m'' is ''T'' → ''T'', also note that the first parameter of ''m'' is this/self) and a derived class type ''S'' from ''T''. By inheritance, the type of ''m'' in ''S'' is ''S'' → ''S''.{{Citation needed|date=July 2022}}  In order for ''S'' to be a subtype of ''T'' the type of ''m'' in ''S'' must be a subtype of the type of ''m'' in ''T'' {{Citation needed|date=July 2022}}, in other words: ''S'' → ''S'' ≤: ''T'' → ''T''. By bottom-up application of the function subtyping rule, this means: ''S'' ≤: ''T'' and ''T'' ≤: ''S'', which is only possible if ''S'' and ''T'' are the same. Since inheritance is an irreflexive relation, ''S'' can't be a subtype of ''T''. 

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 .{{sfn|Cook|Hill|Canning|1990}}

==Coercions==
In coercive subtyping systems, subtypes are defined by explicit [[type conversion]] functions from subtype to supertype. For each subtyping relationship (''S'' <: ''T''), a coercion function ''coerce'': ''S'' → ''T'' is provided, and any object ''s'' of type ''S'' is regarded as the object ''coerce''<sub>''S'' → ''T''</sub>(''s'') of type ''T''. A coercion function may be defined by composition: if ''S'' <: ''T'' and ''T'' <: ''U'' then ''s'' may be regarded as an object of type ''u'' under the compound coercion (''coerce''<sub>''T'' → ''U''</sub> ∘ ''coerce''<sub>''S'' → ''T''</sub>). The [[type coercion]] from a type to itself ''coerce''<sub>''T'' → ''T''</sub> is the [[identity function]] ''id''<sub>T</sub>.

Coercion functions for records and [[disjoint union]] 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 ''f'''(''t'') = ''coerce''<sub>''S''<sub>2</sub> → ''T''<sub>2</sub></sub>(''f''(''coerce''<sub>''T''<sub>1</sub> → ''S''<sub>1</sub></sub>(''t''))), reflecting the [[Covariance and contravariance (computer science)|contravariance]] of parameter values and covariance of return values.

The coercion function is uniquely determined given the subtype and [[supertype]]. 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 : ''int'' can be coerced to a floating point number (say, 2.0 : ''float''), then it is not admissible to coerce 2.1 : ''float'' to 2 : ''int'', because the compound coercion ''coerce''<sub>''float'' → ''float''</sub> given by ''coerce''<sub>''int'' → ''float''</sub> ∘ ''coerce''<sub>''float'' → ''int''</sub> would then be distinct from the identity coercion ''id''<sub>''float''</sub>.

== See also ==
{{wikibooks|Ada Programming|Type System|Subtypes}}
* [[Covariance and contravariance (computer science)|Covariance and contravariance]]
* The [[circle-ellipse problem]] (for the perils of subtyping variable-types on the same basis as value-types)
* [[Class-based programming]]
* [[Top type]]
* [[Refinement type]]
* [[Behavioral subtyping]]

==Notes==
{{reflist}}

== References ==
'''Textbooks'''
{{refbegin}}
* Benjamin C. Pierce, ''Types and programming languages'', MIT Press, 2002, {{ISBN|0-262-16209-1}}, chapter 15 (subtyping of record types), 19.3 (nominal vs. structural types and subtyping), and 23.2 (varieties of polymorphism)
* C. Szyperski, D. Gruntz, S. Murer, ''Component software: beyond object-oriented programming'', 2nd ed., Pearson Education, 2002, {{ISBN|0-201-74572-0}}, pp.&nbsp;93–95 (a high-level presentation aimed at programming language users)
{{refend}}
'''Papers'''
{{refbegin}}{{Cite conference|last1=Cook|first1=William R.|last2=Hill|first2=Walter|last3=Canning|first3=Peter S.|doi=10.1145/96709.96721|title=Inheritance is not subtyping|conference=Proc. 17th ACM SIGPLAN-SIGACT Symp. on Principles of Programming Languages (POPL)|pages=125–135|year=1990|isbn=0-89791-343-4|citeseerx=10.1.1.102.8635}}
* 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 John C. Mitchell, editors, Theoretical Aspects of Object-Oriented Programming: Types, Semantics, and Language Design (MIT Press, 1994).
{{refend}}

== Further reading ==
{{refbegin}}
* [[John C. Reynolds]], ''Theories of programming languages'', Cambridge University Press, 1998, {{ISBN|0-521-59414-6}}, chapter 16.
* [[Martín Abadi]], [[Luca Cardelli]], ''A theory of objects'', Springer, 1996, {{ISBN|0-387-94775-2}}. Section 8.6 contrast the subtyping of records and objects.
{{refend}}

{{data types}}

{{DEFAULTSORT:Subtype Polymorphism}}
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[[Category:Polymorphism (computer science)]]
[[Category:Type theory]]
[[Category:Object-oriented programming]]