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<ul id="toc-TypeScript_example-sublist" class="vector-toc-list"> <li id="toc-Comparison_to_inheritance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Comparison_to_inheritance"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Comparison to inheritance</span> </div> </a> <ul id="toc-Comparison_to_inheritance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Comparison_to_duck_typing" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Comparison_to_duck_typing"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Comparison to duck typing</span> </div> </a> <ul id="toc-Comparison_to_duck_typing-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Comparison_to_function_overloading" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Comparison_to_function_overloading"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Comparison to function overloading</span> </div> </a> <ul id="toc-Comparison_to_function_overloading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Comparison_to_the_visitor_pattern" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Comparison_to_the_visitor_pattern"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Comparison to the visitor pattern</span> </div> </a> <ul id="toc-Comparison_to_the_visitor_pattern-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Limitations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Limitations"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>Limitations</span> </div> </a> <ul id="toc-Limitations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Dependent_intersection_type" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Dependent_intersection_type"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Dependent intersection type</span> </div> </a> <button aria-controls="toc-Dependent_intersection_type-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Dependent intersection type subsection</span> </button> <ul id="toc-Dependent_intersection_type-sublist" class="vector-toc-list"> <li id="toc-Scala_example" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Scala_example"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Scala example</span> </div> </a> <ul id="toc-Scala_example-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Intersection_of_a_type_family" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Intersection_of_a_type_family"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Intersection of a type family</span> </div> </a> <ul id="toc-Intersection_of_a_type_family-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Comparison_of_languages_with_intersection_types" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Comparison_of_languages_with_intersection_types"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Comparison of languages with intersection types</span> </div> </a> <ul id="toc-Comparison_of_languages_with_intersection_types-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet 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.mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar nomobile nowraplinks" style="margin-top: 0.5em;"><tbody><tr><th class="sidebar-title"><a href="/wiki/Type_system" title="Type system">Type systems</a></th></tr><tr><th class="sidebar-heading"> General concepts</th></tr><tr><td class="sidebar-content" style="text-align: left;"> <ul><li><a href="/wiki/Type_safety" title="Type safety">Type safety</a></li> <li><a href="/wiki/Strong_and_weak_typing" title="Strong and weak typing">Strong vs. weak typing</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Major categories</th></tr><tr><td class="sidebar-content" style="text-align: left;"> <ul><li><a href="/wiki/Type_system#Static_type_checking" title="Type system">Static</a> vs. <a href="/wiki/Type_system#Dynamic_type_checking_and_runtime_type_information" title="Type system">dynamic</a></li> <li><a href="/wiki/Manifest_typing" title="Manifest typing">Manifest</a> vs. <a href="/wiki/Type_inference" title="Type inference">inferred</a></li> <li><a href="/wiki/Nominal_type_system" title="Nominal type system">Nominal</a> vs. <a href="/wiki/Structural_type_system" title="Structural type system">structural</a></li> <li><a href="/wiki/Duck_typing" title="Duck typing">Duck typing</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Minor categories</th></tr><tr><td class="sidebar-content" style="text-align: left;"> <ul><li><a href="/wiki/Abstract_type" title="Abstract type">Abstract</a></li> <li><a href="/wiki/Dependent_type" title="Dependent type">Dependent</a></li> <li><a href="/wiki/Flow-sensitive_typing" title="Flow-sensitive typing">Flow-sensitive</a></li> <li><a href="/wiki/Gradual_typing" title="Gradual typing">Gradual</a></li> <li><a class="mw-selflink selflink">Intersection</a></li> <li><a href="/wiki/Latent_typing" title="Latent typing">Latent</a></li> <li><a href="/wiki/Refinement_type" title="Refinement type">Refinement</a></li> <li><a href="/wiki/Substructural_type_system" title="Substructural type system">Substructural</a></li> <li><a href="/wiki/Uniqueness_type" title="Uniqueness type">Unique</a></li> <li><a href="/wiki/Session_type" title="Session type">Session</a></li></ul></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Type_systems" title="Template:Type systems"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Type_systems" title="Template talk:Type systems"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Type_systems" title="Special:EditPage/Template:Type systems"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Type_theory" title="Type theory">type theory</a>, an <b>intersection type</b> can be allocated to values that can be assigned both the type <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" /></span> and the type <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" /></span>. This value can be given the intersection type <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma \cap \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>∩</mo> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma \cap \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1b5fbc85a391ea8ded8940802c28265a131830b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.114ex; height:2.009ex;" alt="{\displaystyle \sigma \cap \tau }" /></span> in an <a href="/wiki/Intersection_type_system" class="mw-redirect" title="Intersection type system">intersection type system</a>.<sup id="cite_ref-BCD83_1-0" class="reference"><a href="#cite_note-BCD83-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Generally, if the ranges of values of two types overlap, then a value belonging to the <i>intersection</i> of the two ranges can be assigned the <i>intersection type</i> of these two types. Such a value can be safely passed as argument to functions expecting <i>either</i> of the two types. For example, in <a href="/wiki/Java_(programming_language)" title="Java (programming language)">Java</a> the class <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">Boolean</code> implements both the <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">Serializable</code> and the <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">Comparable</code> interfaces. Therefore, an object of type <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">Boolean</code> can be safely passed to functions expecting an argument of type <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">Serializable</code> and to functions expecting an argument of type <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">Comparable</code>. </p><p>Intersection types are <a href="/wiki/Composite_data_type" title="Composite data type">composite data types</a>. Similar to <a href="/wiki/Product_type" title="Product type">product types</a>, they are used to assign several types to an object. However, product types are assigned to <a href="/wiki/Tuple" title="Tuple">tuples</a>, so that each tuple element is assigned a particular product type component. In comparison, underlying objects of intersection types are not necessarily composite. A restricted form of intersection types are <a href="/wiki/Refinement_type" title="Refinement type">refinement types</a>. </p><p>Intersection types are useful for describing <a href="/wiki/Function_overloading" title="Function overloading">overloaded functions</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> For example, if <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="kt">number</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="kt">number</span></code> is the type of function taking a number as an argument and returning a number, and <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="kt">string</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="kt">string</span></code> is the type of function taking a string as an argument and returning a string, then the intersection of these two types can be used to describe (overloaded) functions that do one or the other, based on what type of input they are given. </p><p>Contemporary programming languages, including <a href="/wiki/Ceylon_(programming_language)" title="Ceylon (programming language)">Ceylon</a>, Flow, <a href="/wiki/Java_(programming_language)" title="Java (programming language)">Java</a>, <a href="/wiki/Scala_(programming_language)" title="Scala (programming language)">Scala</a>, <a href="/wiki/Microsoft_TypeScript" class="mw-redirect" title="Microsoft TypeScript">TypeScript</a>, and <a href="/wiki/Whiley_(programming_language)" title="Whiley (programming language)">Whiley</a> (see <a href="#Comparison_of_languages_with_intersection_types">comparison of languages with intersection types</a>), use intersection types to combine interface specifications and to express <a href="/wiki/Ad_hoc_polymorphism" title="Ad hoc polymorphism">ad hoc polymorphism</a>. Complementing <a href="/wiki/Parametric_polymorphism" title="Parametric polymorphism">parametric polymorphism</a>, intersection types may be used to avoid class hierarchy pollution from <a href="/wiki/Cross-cutting_concern" title="Cross-cutting concern">cross-cutting concerns</a> and reduce <a href="/wiki/Boilerplate_code" title="Boilerplate code">boilerplate code</a>, as shown in the <a href="#TypeScript_example">TypeScript example</a> below. </p><p>The <a href="/wiki/Type_theory" title="Type theory">type theoretic</a> study of intersection types is referred to as the <a href="/wiki/Intersection_type_discipline" title="Intersection type discipline">intersection type discipline</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Remarkably, program termination can be precisely characterized using intersection types.<sup id="cite_ref-G96_4-0" class="reference"><a href="#cite_note-G96-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="TypeScript_example">TypeScript example</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Intersection_type&action=edit&section=1" title="Edit section: TypeScript example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/TypeScript" title="TypeScript">TypeScript</a> supports intersection types,<sup id="cite_ref-TS_5-0" class="reference"><a href="#cite_note-TS-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> improving expressiveness of the type system and reducing potential class hierarchy size, demonstrated as follows. </p><p>The following program code defines the classes <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">Chicken</span></code>, <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">Cow</span></code>, and <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">RandomNumberGenerator</span></code> that each have a method <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">produce</span></code> returning an object of either type <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">Egg</span></code>, <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">Milk</span></code>, or <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="kt">number</span></code>. Additionally, the functions <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">eatEgg</span></code> and <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">drinkMilk</span></code> require arguments of type <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">Egg</span></code> and <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">Milk</span></code>, respectively. </p> <div class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><pre><span></span><span class="kd">class</span><span class="w"> </span><span class="nx">Egg</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="k">private</span><span class="w"> </span><span class="nx">kind</span><span class="o">:</span><span class="w"> </span><span class="s2">"Egg"</span><span class="w"> </span><span class="p">}</span> <span class="kd">class</span><span class="w"> </span><span class="nx">Milk</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="k">private</span><span class="w"> </span><span class="nx">kind</span><span class="o">:</span><span class="w"> </span><span class="s2">"Milk"</span><span class="w"> </span><span class="p">}</span> <span class="c1">// produces eggs</span> <span class="kd">class</span><span class="w"> </span><span class="nx">Chicken</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="nx">produce</span><span class="p">()</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">Egg</span><span class="p">();</span><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="p">}</span> <span class="c1">// produces milk</span> <span class="kd">class</span><span class="w"> </span><span class="nx">Cow</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="nx">produce</span><span class="p">()</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">Milk</span><span class="p">();</span><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="p">}</span> <span class="c1">// produces a random number</span> <span class="kd">class</span><span class="w"> </span><span class="nx">RandomNumberGenerator</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="nx">produce</span><span class="p">()</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">random</span><span class="p">();</span><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="p">}</span> <span class="c1">// requires an egg</span> <span class="kd">function</span><span class="w"> </span><span class="nx">eatEgg</span><span class="p">(</span><span class="nx">egg</span><span class="o">:</span><span class="w"> </span><span class="kt">Egg</span><span class="p">)</span><span class="w"> </span><span class="p">{</span> <span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="s2">"I ate an egg."</span><span class="p">;</span> <span class="p">}</span> <span class="c1">// requires milk</span> <span class="kd">function</span><span class="w"> </span><span class="nx">drinkMilk</span><span class="p">(</span><span class="nx">milk</span><span class="o">:</span><span class="w"> </span><span class="kt">Milk</span><span class="p">)</span><span class="w"> </span><span class="p">{</span> <span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="s2">"I drank some milk."</span><span class="p">;</span> <span class="p">}</span> </pre></div> <p>The following program code defines the <a href="/wiki/Ad_hoc_polymorphism" title="Ad hoc polymorphism">ad hoc polymorphic</a> function <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">animalToFood</span></code> that invokes the member function <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">produce</span></code> of the given object <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">animal</span></code>. The function <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">animalToFood</span></code> has <i>two</i> type annotations, namely <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="p">((</span><span class="nx">_</span><span class="o">:</span><span class="w"> </span><span class="kt">Chicken</span><span class="p">)</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="nx">Egg</span><span class="p">)</span></code> and <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="p">((</span><span class="nx">_</span><span class="o">:</span><span class="w"> </span><span class="kt">Cow</span><span class="p">)</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="nx">Milk</span><span class="p">)</span></code>, connected via the intersection type constructor <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="o">&</span></code>. Specifically, <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">animalToFood</span></code> when applied to an argument of type <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">Chicken</span></code> returns an object of type <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">Egg</span></code>, and when applied to an argument of type <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">Cow</span></code> returns an object of type <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">Milk</span></code>. Ideally, <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">animalToFood</span></code> should not be applicable to any object having (possibly by chance) a <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">produce</span></code> method. </p> <div class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><pre><span></span><span class="c1">// given a chicken, produces an egg; given a cow, produces milk</span> <span class="kd">let</span><span class="w"> </span><span class="nx">animalToFood</span><span class="o">:</span><span class="w"> </span><span class="p">((</span><span class="nx">_</span><span class="o">:</span><span class="w"> </span><span class="kt">Chicken</span><span class="p">)</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="nx">Egg</span><span class="p">)</span><span class="w"> </span><span class="o">&</span><span class="w"> </span><span class="p">((</span><span class="nx">_</span><span class="o">:</span><span class="w"> </span><span class="kt">Cow</span><span class="p">)</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="nx">Milk</span><span class="p">)</span><span class="w"> </span><span class="o">=</span> <span class="w"> </span><span class="kd">function</span><span class="w"> </span><span class="p">(</span><span class="nx">animal</span><span class="o">:</span><span class="w"> </span><span class="kt">any</span><span class="p">)</span><span class="w"> </span><span class="p">{</span> <span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">animal</span><span class="p">.</span><span class="nx">produce</span><span class="p">();</span> <span class="w"> </span><span class="p">};</span> </pre></div> <p>Finally, the following program code demonstrates <a href="/wiki/Type_safety" title="Type safety">type safe</a> use of the above definitions. </p> <div class="mw-highlight mw-highlight-lang-typescript mw-content-ltr mw-highlight-lines" dir="ltr"><pre><span></span><span class="linenos" data-line="1"></span><span class="kd">var</span><span class="w"> </span><span class="nx">chicken</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">Chicken</span><span class="p">();</span> <span class="linenos" data-line="2"></span><span class="kd">var</span><span class="w"> </span><span class="nx">cow</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">Cow</span><span class="p">();</span> <span class="linenos" data-line="3"></span><span class="kd">var</span><span class="w"> </span><span class="nx">randomNumberGenerator</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">RandomNumberGenerator</span><span class="p">();</span> <span class="linenos" data-line="4"></span> <span class="linenos" data-line="5"></span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="nx">chicken</span><span class="p">.</span><span class="nx">produce</span><span class="p">());</span><span class="w"> </span><span class="c1">// Egg { }</span> <span class="linenos" data-line="6"></span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="nx">cow</span><span class="p">.</span><span class="nx">produce</span><span class="p">());</span><span class="w"> </span><span class="c1">// Milk { }</span> <span class="linenos" data-line="7"></span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="nx">randomNumberGenerator</span><span class="p">.</span><span class="nx">produce</span><span class="p">());</span><span class="w"> </span><span class="c1">//0.2626353555444987</span> <span class="linenos" data-line="8"></span> <span class="linenos" data-line="9"></span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="nx">animalToFood</span><span class="p">(</span><span class="nx">chicken</span><span class="p">));</span><span class="w"> </span><span class="c1">// Egg { }</span> <span class="linenos" data-line="10"></span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="nx">animalToFood</span><span class="p">(</span><span class="nx">cow</span><span class="p">));</span><span class="w"> </span><span class="c1">// Milk { }</span> <span class="linenos" data-line="11"></span><span class="c1">//console.log(animalToFood(randomNumberGenerator)); // ERROR: Argument of type 'RandomNumberGenerator' is not assignable to parameter of type 'Cow'</span> <span class="linenos" data-line="12"></span> <span class="linenos" data-line="13"></span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="nx">eatEgg</span><span class="p">(</span><span class="nx">animalToFood</span><span class="p">(</span><span class="nx">chicken</span><span class="p">)));</span><span class="w"> </span><span class="c1">// I ate an egg.</span> <span class="linenos" data-line="14"></span><span class="c1">//console.log(eatEgg(animalToFood(cow))); // ERROR: Argument of type 'Milk' is not assignable to parameter of type 'Egg'</span> <span class="linenos" data-line="15"></span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="nx">drinkMilk</span><span class="p">(</span><span class="nx">animalToFood</span><span class="p">(</span><span class="nx">cow</span><span class="p">)));</span><span class="w"> </span><span class="c1">// I drank some milk.</span> <span class="linenos" data-line="16"></span><span class="c1">//console.log(drinkMilk(animalToFood(chicken))); // ERROR: Argument of type 'Egg' is not assignable to parameter of type 'Milk'</span> </pre></div> <p>The above program code has the following properties: </p> <ul><li>Lines 1–3 create objects <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">chicken</span></code>, <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">cow</span></code>, and <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">randomNumberGenerator</span></code> of their respective type.</li> <li>Lines 5–7 print for the previously created objects the respective results (provided as comments) when invoking <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">produce</span></code>.</li> <li>Line 9 (resp. 10) demonstrates type safe use of the method <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">animalToFood</span></code> applied to <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">chicken</span></code> (resp. <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">cow</span></code>).</li> <li>Line 11, if uncommented, would result in a type error at compile time. Although the <i>implementation</i> of <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">animalToFood</span></code> could invoke the <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">produce</span></code> method of <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">randomNumberGenerator</span></code>, the <i>type annotation</i> of <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">animalToFood</span></code> disallows it. This is in accordance with the intended meaning of <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">animalToFood</span></code>.</li> <li>Line 13 (resp. 15) demonstrates that applying <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">animalToFood</span></code> to <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">chicken</span></code> (resp. <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">cow</span></code>) results in an object of type <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">Egg</span></code> (resp. <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">Milk</span></code>).</li> <li>Line 14 (resp. 16) demonstrates that applying <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">animalToFood</span></code> to <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">cow</span></code> (resp. <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">chicken</span></code>) does not result in an object of type <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">Egg</span></code> (resp. <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">Milk</span></code>). Therefore, if uncommented, line 14 (resp. 16) would result in a type error at compile time.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Comparison_to_inheritance">Comparison to inheritance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Intersection_type&action=edit&section=2" title="Edit section: Comparison to inheritance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The above minimalist example can be realized using <a href="/wiki/Inheritance_(object-oriented_programming)" title="Inheritance (object-oriented programming)">inheritance</a>, for instance by deriving the classes <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">Chicken</span></code> and <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">Cow</span></code> from a base class <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">Animal</span></code>. However, in a larger setting, this could be disadvantageous. Introducing new classes into a class hierarchy is not necessarily justified for <a href="/wiki/Cross-cutting_concern" title="Cross-cutting concern">cross-cutting concerns</a>, or maybe outright impossible, for example when using an external library. Imaginably, the above example could be extended with the following classes: </p> <ul><li>a class <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">Horse</span></code> that does not have a <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">produce</span></code> method;</li> <li>a class <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">Sheep</span></code> that has a <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">produce</span></code> method returning <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">Wool</span></code>;</li> <li>a class <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">Pig</span></code> that has a <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">produce</span></code> method, which can be used only once, returning <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">Meat</span></code>.</li></ul> <p>This may require additional classes (or interfaces) specifying whether a produce method is available, whether the produce method returns food, and whether the produce method can be used repeatedly. Overall, this may pollute the class hierarchy. </p> <div class="mw-heading mw-heading3"><h3 id="Comparison_to_duck_typing">Comparison to duck typing</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Intersection_type&action=edit&section=3" title="Edit section: Comparison to duck typing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The above minimalist example already shows that <a href="/wiki/Duck_typing" title="Duck typing">duck typing</a> is less suited to realize the given scenario. While the class <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">RandomNumberGenerator</span></code> contains a <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">produce</span></code> method, the object <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">randomNumberGenerator</span></code> should not be a valid argument for <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">animalToFood</span></code>. The above example can be realized using duck typing, for instance by introducing a new field <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">argumentForAnimalToFood</span></code> to the classes <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">Chicken</span></code> and <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">Cow</span></code> signifying that objects of corresponding type are valid arguments for <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">animalToFood</span></code>. However, this would not only increase the size of the respective classes (especially with the introduction of more methods similar to <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">animalToFood</span></code>), but is also a non-local approach with respect to <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">animalToFood</span></code>. </p> <div class="mw-heading mw-heading3"><h3 id="Comparison_to_function_overloading">Comparison to function overloading</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Intersection_type&action=edit&section=4" title="Edit section: Comparison to function overloading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The above example can be realized using <a href="/wiki/Function_overloading" title="Function overloading">function overloading</a>, for instance by implementing two methods <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">animalToFood</span><span class="p">(</span><span class="nx">animal</span><span class="o">:</span><span class="w"> </span><span class="kt">Chicken</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="nx">Egg</span></code> and <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">animalToFood</span><span class="p">(</span><span class="nx">animal</span><span class="o">:</span><span class="w"> </span><span class="kt">Cow</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="nx">Milk</span></code>. In TypeScript, such a solution is almost identical to the provided example. Other programming languages, such as <a href="/wiki/Java_(programming_language)" title="Java (programming language)">Java</a>, require distinct implementations of the overloaded method. This may lead to either <a href="/wiki/Code_duplication" class="mw-redirect" title="Code duplication">code duplication</a> or <a href="/wiki/Boilerplate_code" title="Boilerplate code">boilerplate code</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Comparison_to_the_visitor_pattern">Comparison to the visitor pattern</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Intersection_type&action=edit&section=5" title="Edit section: Comparison to the visitor pattern"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The above example can be realized using the <a href="/wiki/Visitor_pattern" title="Visitor pattern">visitor pattern</a>. It would require each animal class to implement an <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">accept</span></code> method accepting an object implementing the interface <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">AnimalVisitor</span></code> (adding non-local <a href="/wiki/Boilerplate_code" title="Boilerplate code">boilerplate code</a>). The function <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">animalToFood</span></code> would be realized as the <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">visit</span></code> method of an implementation of <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">AnimalVisitor</span></code>. Unfortunately, the connection between the input type (<code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">Chicken</span></code> or <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">Cow</span></code>) and the result type (<code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">Egg</span></code> or <code class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><span class="nx">Milk</span></code>) would be difficult to represent. </p> <div class="mw-heading mw-heading3"><h3 id="Limitations">Limitations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Intersection_type&action=edit&section=6" title="Edit section: Limitations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>On the one hand, intersection types <i>can</i> be used to locally annotate different types to a function without introducing new classes (or interfaces) to the class hierarchy. On the other hand, this approach <i>requires</i> all possible argument types and result types to be specified explicitly. If the behavior of a function can be specified precisely by either a unified interface, <a href="/wiki/Parametric_polymorphism" title="Parametric polymorphism">parametric polymorphism</a>, or <a href="/wiki/Duck_typing" title="Duck typing">duck typing</a>, then the verbose nature of intersection types is unfavorable. Therefore, intersection types should be considered complementary to existing specification methods. </p> <div class="mw-heading mw-heading2"><h2 id="Dependent_intersection_type">Dependent intersection type</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Intersection_type&action=edit&section=7" title="Edit section: Dependent intersection type"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <b>dependent intersection type</b>, denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x:\sigma )\cap \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>:</mo> <mi>σ</mi> <mo stretchy="false">)</mo> <mo>∩</mo> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x:\sigma )\cap \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d1582886a9278c7fc635dac09779d9546f7a372" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.19ex; height:2.843ex;" alt="{\displaystyle (x:\sigma )\cap \tau }" /></span>, is a <a href="/wiki/Dependent_type" title="Dependent type">dependent type</a> in which the type <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" /></span> may depend on the term variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /></span>.<sup id="cite_ref-K03_6-0" class="reference"><a href="#cite_note-K03-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> In particular, if a term <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span> has the dependent intersection type <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x:\sigma )\cap \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>:</mo> <mi>σ</mi> <mo stretchy="false">)</mo> <mo>∩</mo> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x:\sigma )\cap \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d1582886a9278c7fc635dac09779d9546f7a372" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.19ex; height:2.843ex;" alt="{\displaystyle (x:\sigma )\cap \tau }" /></span>, then the term <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span> has <i>both</i> the type <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" /></span> and the type <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau [x:=M]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo>:=</mo> <mi>M</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau [x:=M]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd0c8300581da4d7f300aa14361be64c101020cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.013ex; height:2.843ex;" alt="{\displaystyle \tau [x:=M]}" /></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau [x:=M]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo>:=</mo> <mi>M</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau [x:=M]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd0c8300581da4d7f300aa14361be64c101020cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.013ex; height:2.843ex;" alt="{\displaystyle \tau [x:=M]}" /></span> is the type which results from replacing all occurrences of the term variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" /></span> by the term <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Scala_example">Scala example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Intersection_type&action=edit&section=8" title="Edit section: Scala example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Scala_(programming_language)" title="Scala (programming language)">Scala</a> supports type declarations <sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> as object members. This allows a type of an object member to depend on the value of another member, which is called a <i>path-dependent type</i>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> For example, the following program text defines a Scala trait <code class="mw-highlight mw-highlight-lang-scala mw-content-ltr" dir="ltr"><span class="nc">Witness</span></code>, which can be used to implement the <a href="/wiki/Singleton_pattern" title="Singleton pattern">singleton pattern</a>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-highlight mw-highlight-lang-scala mw-content-ltr" dir="ltr"><pre><span></span><span class="k">trait</span><span class="w"> </span><span class="nc">Witness</span><span class="w"> </span><span class="p">{</span> <span class="w"> </span><span class="k">type</span><span class="w"> </span><span class="nc">T</span> <span class="w"> </span><span class="kd">val</span><span class="w"> </span><span class="n">value</span><span class="p">:</span><span class="w"> </span><span class="nc">T</span><span class="w"> </span><span class="p">{}</span> <span class="p">}</span> </pre></div> <p>The above trait <code class="mw-highlight mw-highlight-lang-scala mw-content-ltr" dir="ltr"><span class="nc">Witness</span></code> declares the member <code class="mw-highlight mw-highlight-lang-scala mw-content-ltr" dir="ltr"><span class="nc">T</span></code>, which can be assigned a <i>type</i> as its value, and the member <code class="mw-highlight mw-highlight-lang-scala mw-content-ltr" dir="ltr"><span class="n">value</span></code>, which can be assigned a value of type <code class="mw-highlight mw-highlight-lang-scala mw-content-ltr" dir="ltr"><span class="nc">T</span></code>. The following program text defines an object <code class="mw-highlight mw-highlight-lang-scala mw-content-ltr" dir="ltr"><span class="n">booleanWitness</span></code> as instance of the above trait <code class="mw-highlight mw-highlight-lang-scala mw-content-ltr" dir="ltr"><span class="nc">Witness</span></code>. The object <code class="mw-highlight mw-highlight-lang-scala mw-content-ltr" dir="ltr"><span class="n">booleanWitness</span></code> defines the type <code class="mw-highlight mw-highlight-lang-scala mw-content-ltr" dir="ltr"><span class="nc">T</span></code> as <code class="mw-highlight mw-highlight-lang-scala mw-content-ltr" dir="ltr"><span class="nc">Boolean</span></code> and the value <code class="mw-highlight mw-highlight-lang-scala mw-content-ltr" dir="ltr"><span class="n">value</span></code> as <code class="mw-highlight mw-highlight-lang-scala mw-content-ltr" dir="ltr"><span class="kc">true</span></code>. For example, executing <code class="mw-highlight mw-highlight-lang-scala mw-content-ltr" dir="ltr"><span class="nc">System</span><span class="p">.</span><span class="n">out</span><span class="p">.</span><span class="n">println</span><span class="p">(</span><span class="n">booleanWitness</span><span class="p">.</span><span class="n">value</span><span class="p">)</span></code> prints <code class="mw-highlight mw-highlight-lang-scala mw-content-ltr" dir="ltr"><span class="kc">true</span></code> on the console. </p> <div class="mw-highlight mw-highlight-lang-scala mw-content-ltr" dir="ltr"><pre><span></span><span class="k">object</span><span class="w"> </span><span class="nc">booleanWitness</span><span class="w"> </span><span class="k">extends</span><span class="w"> </span><span class="nc">Witness</span><span class="w"> </span><span class="p">{</span> <span class="w"> </span><span class="k">type</span><span class="w"> </span><span class="nc">T</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nc">Boolean</span> <span class="w"> </span><span class="kd">val</span><span class="w"> </span><span class="n">value</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span> <span class="p">}</span> </pre></div> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle {\textsf {x}}:\sigma \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">x</mtext> </mrow> </mrow> <mo>:</mo> <mi>σ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle {\textsf {x}}:\sigma \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2b03d3c8d2be2e064edb383d050bc5aebda4368" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.148ex; height:2.843ex;" alt="{\displaystyle \langle {\textsf {x}}:\sigma \rangle }" /></span> be the type (specifically, a <a href="/wiki/Record_(computer_science)" title="Record (computer science)">record type</a>) of objects having the member <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textsf {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">x</mtext> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textsf {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0350688214b6c2cbd3a985ce656f08ad9660372b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.072ex; height:1.676ex;" alt="{\displaystyle {\textsf {x}}}" /></span> of type <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" /></span>. In the above example, the object <code class="mw-highlight mw-highlight-lang-scala mw-content-ltr" dir="ltr"><span class="n">booleanWitness</span></code> can be assigned the dependent intersection type <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x:\langle {\textsf {T}}:{\text{Type}}\rangle )\cap \langle {\textsf {value}}:x.{\textsf {T}}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>:</mo> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Type</mtext> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo stretchy="false">)</mo> <mo>∩</mo> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">value</mtext> </mrow> </mrow> <mo>:</mo> <mi>x</mi> <mo>.</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x:\langle {\textsf {T}}:{\text{Type}}\rangle )\cap \langle {\textsf {value}}:x.{\textsf {T}}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35db40712cb5b22bdc1973875c2163541d7dcaea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.892ex; height:2.843ex;" alt="{\displaystyle (x:\langle {\textsf {T}}:{\text{Type}}\rangle )\cap \langle {\textsf {value}}:x.{\textsf {T}}\rangle }" /></span>. The reasoning is as follows. The object <code class="mw-highlight mw-highlight-lang-scala mw-content-ltr" dir="ltr"><span class="n">booleanWitness</span></code> has the member <code class="mw-highlight mw-highlight-lang-scala mw-content-ltr" dir="ltr"><span class="nc">T</span></code> that is assigned the type <code class="mw-highlight mw-highlight-lang-scala mw-content-ltr" dir="ltr"><span class="nc">Boolean</span></code> as its value. Since <code class="mw-highlight mw-highlight-lang-scala mw-content-ltr" dir="ltr"><span class="nc">Boolean</span></code> is a type, the object <code class="mw-highlight mw-highlight-lang-scala mw-content-ltr" dir="ltr"><span class="n">booleanWitness</span></code> has the type <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle {\textsf {T}}:{\text{Type}}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Type</mtext> </mrow> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle {\textsf {T}}:{\text{Type}}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09a4e39ede9bdbd8aa3b62a9fb8eae947f680b68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.56ex; height:2.843ex;" alt="{\displaystyle \langle {\textsf {T}}:{\text{Type}}\rangle }" /></span>. Additionally, the object <code class="mw-highlight mw-highlight-lang-scala mw-content-ltr" dir="ltr"><span class="n">booleanWitness</span></code> has the member <code class="mw-highlight mw-highlight-lang-scala mw-content-ltr" dir="ltr"><span class="n">value</span></code> that is assigned the value <code class="mw-highlight mw-highlight-lang-scala mw-content-ltr" dir="ltr"><span class="kc">true</span></code> of type <code class="mw-highlight mw-highlight-lang-scala mw-content-ltr" dir="ltr"><span class="nc">Boolean</span></code>. Since the value of <code class="mw-highlight mw-highlight-lang-scala mw-content-ltr" dir="ltr"><span class="n">booleanWitness</span><span class="p">.</span><span class="nc">T</span></code> is <code class="mw-highlight mw-highlight-lang-scala mw-content-ltr" dir="ltr"><span class="nc">Boolean</span></code>, the object <code class="mw-highlight mw-highlight-lang-scala mw-content-ltr" dir="ltr"><span class="n">booleanWitness</span></code> has the type <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle {\textsf {value}}:{\textsf {booleanWitness.T}}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">value</mtext> </mrow> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">booleanWitness.T</mtext> </mrow> </mrow> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle {\textsf {value}}:{\textsf {booleanWitness.T}}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30afb173f0c6589ca655fdfe34fda78c887a220e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.998ex; height:2.843ex;" alt="{\displaystyle \langle {\textsf {value}}:{\textsf {booleanWitness.T}}\rangle }" /></span>. Overall, the object <code class="mw-highlight mw-highlight-lang-scala mw-content-ltr" dir="ltr"><span class="n">booleanWitness</span></code> has the intersection type <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle {\textsf {T}}:{\text{Type}}\rangle \cap \langle {\textsf {value}}:{\textsf {booleanWitness.T}}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Type</mtext> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>∩</mo> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">value</mtext> </mrow> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">booleanWitness.T</mtext> </mrow> </mrow> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle {\textsf {T}}:{\text{Type}}\rangle \cap \langle {\textsf {value}}:{\textsf {booleanWitness.T}}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b01633bcb1c5f27b635e7dbf922c9b0fdc89f95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.14ex; height:2.843ex;" alt="{\displaystyle \langle {\textsf {T}}:{\text{Type}}\rangle \cap \langle {\textsf {value}}:{\textsf {booleanWitness.T}}\rangle }" /></span>. Therefore, presenting self-reference as dependency, the object <code class="mw-highlight mw-highlight-lang-scala mw-content-ltr" dir="ltr"><span class="n">booleanWitness</span></code> has the dependent intersection type <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x:\langle {\textsf {T}}:{\text{Type}}\rangle )\cap \langle {\textsf {value}}:x.{\textsf {T}}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>:</mo> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Type</mtext> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo stretchy="false">)</mo> <mo>∩</mo> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">value</mtext> </mrow> </mrow> <mo>:</mo> <mi>x</mi> <mo>.</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x:\langle {\textsf {T}}:{\text{Type}}\rangle )\cap \langle {\textsf {value}}:x.{\textsf {T}}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35db40712cb5b22bdc1973875c2163541d7dcaea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.892ex; height:2.843ex;" alt="{\displaystyle (x:\langle {\textsf {T}}:{\text{Type}}\rangle )\cap \langle {\textsf {value}}:x.{\textsf {T}}\rangle }" /></span>. </p><p>Alternatively, the above minimalistic example can be described using <i>dependent record types</i>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> In comparison to dependent intersection types, dependent record types constitute a strictly more specialized type theoretic concept.<sup id="cite_ref-K03_6-1" class="reference"><a href="#cite_note-K03-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Intersection_of_a_type_family">Intersection of a type family</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Intersection_type&action=edit&section=9" title="Edit section: Intersection of a type family"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An <b>intersection of a type family</b>, denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \bigcap _{x:\sigma }\tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>:</mo> <mi>σ</mi> </mrow> </munder> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \bigcap _{x:\sigma }\tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44efcaf5df3a92bb171a16211e44f33e85cc94ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.095ex; height:3.009ex;" alt="{\textstyle \bigcap _{x:\sigma }\tau }" /></span>, is a <a href="/wiki/Dependent_type" title="Dependent type">dependent type</a> in which the type <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" /></span> may depend on the term variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /></span>. In particular, if a term <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span> has the type <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \bigcap _{x:\sigma }\tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>:</mo> <mi>σ</mi> </mrow> </munder> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \bigcap _{x:\sigma }\tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44efcaf5df3a92bb171a16211e44f33e85cc94ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.095ex; height:3.009ex;" alt="{\textstyle \bigcap _{x:\sigma }\tau }" /></span>, then for <i>each</i> term <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" /></span> of type <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" /></span>, the term <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span> has the type <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau [x:=N]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo>:=</mo> <mi>N</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau [x:=N]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c002d47c768eee1680120285b350790d373dbf23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.634ex; height:2.843ex;" alt="{\displaystyle \tau [x:=N]}" /></span>. This notion is also called <i>implicit <a href="/wiki/Dependent_type#Pi_type" title="Dependent type">Pi type</a></i>,<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> observing that the argument <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" /></span> is not kept at term level. </p> <div class="mw-heading mw-heading2"><h2 id="Comparison_of_languages_with_intersection_types">Comparison of languages with intersection types</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Intersection_type&action=edit&section=10" title="Edit section: Comparison of languages with intersection types"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable sortable"> <tbody><tr> <th>Language</th> <th>Actively developed</th> <th>Paradigm(s)</th> <th>Status</th> <th>Features </th></tr> <tr> <td><a href="/wiki/C_Sharp_(programming_language)" title="C Sharp (programming language)">C#</a></td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup></td> <td> <ul><li><a href="/wiki/Functional_programming" title="Functional programming">Functional</a></li> <li><a href="/wiki/Imperative_programming" title="Imperative programming">Imperative</a></li> <li><a href="/wiki/Object-oriented_programming" title="Object-oriented programming">Object-oriented</a></li></ul> </td> <td style="background: #FF8; color:black; vertical-align: middle; text-align: center;" class="table-maybe">Under discussion<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></td> <td>Additionally, generic type parameters can have constraints that require their (monomorphized) type-arguments to implement multiple interfaces, whereupon the runtime type represented by the generic type parameter becomes an intersection-type of all listed interfaces. </td></tr> <tr> <td><a href="/wiki/Ceylon_(programming_language)" title="Ceylon (programming language)">Ceylon</a></td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup></td> <td> <ul><li><a href="/wiki/Object-oriented_programming" title="Object-oriented programming">Object-oriented</a></li></ul> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Supported<sup id="cite_ref-Cey_15-0" class="reference"><a href="#cite_note-Cey-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup></td> <td> <ul><li>Type refinement</li> <li>Interface composition</li> <li>Subtyping in width</li></ul> </td></tr> <tr> <td><a href="/wiki/F_Sharp_(programming_language)" title="F Sharp (programming language)">F#</a></td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup></td> <td> <ul><li><a href="/wiki/Functional_programming" title="Functional programming">Functional</a></li> <li><a href="/wiki/Imperative_programming" title="Imperative programming">Imperative</a></li> <li><a href="/wiki/Object-oriented_programming" title="Object-oriented programming">Object-oriented</a></li></ul> </td> <td style="background: #FF8; color:black; vertical-align: middle; text-align: center;" class="table-maybe">Under discussion<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup></td> <td style="background: var(--background-color-interactive, #EEE); color: var(--color-base, black); vertical-align: middle; white-space: nowrap; text-align: center;" class="table-Unknown">? </td></tr> <tr> <td>Flow</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup></td> <td> <ul><li><a href="/wiki/Functional_programming" title="Functional programming">Functional</a></li> <li><a href="/wiki/Imperative_programming" title="Imperative programming">Imperative</a></li> <li><a href="/wiki/Object-oriented_programming" title="Object-oriented programming">Object-oriented</a></li> <li><a href="/wiki/Scripting_language" title="Scripting language">Scripting</a></li></ul> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Supported<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup></td> <td> <ul><li>Type refinement</li> <li>Interface composition</li></ul> </td></tr> <tr> <td><a href="/wiki/Forsythe_(programming_language)" class="mw-redirect" title="Forsythe (programming language)">Forsythe</a></td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No</td> <td> <ul><li><a href="/wiki/Imperative_programming" title="Imperative programming">Imperative</a></li></ul> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Supported<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </td> <td> <ul><li>Function type intersection</li> <li>Distributive, co- and contravariant function type subtyping</li></ul> </td></tr> <tr> <td><a href="/wiki/Java_(programming_language)" title="Java (programming language)">Java</a></td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </td> <td> <ul><li><a href="/wiki/Imperative_programming" title="Imperative programming">Imperative</a></li> <li><a href="/wiki/Object-oriented_programming" title="Object-oriented programming">Object-oriented</a></li></ul> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Supported<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </td> <td> <ul><li>Type refinement</li> <li>Interface composition</li> <li>Subtyping in width</li></ul> </td></tr> <tr> <td><a href="/wiki/PHP" title="PHP">PHP</a></td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </td> <td> <ul><li><a href="/wiki/Functional_programming" title="Functional programming">Functional</a></li> <li><a href="/wiki/Imperative_programming" title="Imperative programming">Imperative</a></li> <li><a href="/wiki/Object-oriented_programming" title="Object-oriented programming">Object-oriented</a></li> <li><a href="/wiki/Scripting_language" title="Scripting language">Scripting</a></li></ul> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Supported<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </td> <td> <ul><li>Type refinement</li> <li>Interface composition</li></ul> </td></tr> <tr> <td><a href="/wiki/Scala_(programming_language)" title="Scala (programming language)">Scala</a></td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </td> <td> <ul><li><a href="/wiki/Functional_programming" title="Functional programming">Functional</a></li> <li><a href="/wiki/Imperative_programming" title="Imperative programming">Imperative</a></li> <li><a href="/wiki/Object-oriented_programming" title="Object-oriented programming">Object-oriented</a></li></ul> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Supported<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </td> <td> <ul><li>Type refinement</li> <li>Trait composition</li> <li>Subtyping in width</li></ul> </td></tr> <tr> <td><a href="/wiki/Microsoft_TypeScript" class="mw-redirect" title="Microsoft TypeScript">TypeScript</a></td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup></td> <td> <ul><li><a href="/wiki/Functional_programming" title="Functional programming">Functional</a></li> <li><a href="/wiki/Imperative_programming" title="Imperative programming">Imperative</a></li> <li><a href="/wiki/Object-oriented_programming" title="Object-oriented programming">Object-oriented</a></li> <li><a href="/wiki/Scripting_language" title="Scripting language">Scripting</a></li></ul> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Supported<sup id="cite_ref-TS_5-1" class="reference"><a href="#cite_note-TS-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></td> <td> <ul><li>Arbitrary type intersection</li> <li>Interface composition</li> <li>Subtyping in width and depth</li></ul> </td></tr> <tr> <td><a href="/wiki/Whiley_(programming_language)" title="Whiley (programming language)">Whiley</a></td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup></td> <td> <ul><li><a href="/wiki/Functional_programming" title="Functional programming">Functional</a></li> <li><a href="/wiki/Imperative_programming" title="Imperative programming">Imperative</a></li></ul> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Supported<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup></td> <td style="background: var(--background-color-interactive, #EEE); color: var(--color-base, black); vertical-align: middle; white-space: nowrap; text-align: center;" class="table-Unknown">? </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Intersection_type&action=edit&section=11" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-BCD83-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-BCD83_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBarendregtCoppoDezani-Ciancaglini1983" class="citation journal cs1">Barendregt, Henk; Coppo, Mario; <a href="/wiki/Mariangiola_Dezani-Ciancaglini" title="Mariangiola Dezani-Ciancaglini">Dezani-Ciancaglini, Mariangiola</a> (1983). "A filter lambda model and the completeness of type assignment". <i>Journal of Symbolic Logic</i>. <b>48</b> (4): <span class="nowrap">931–</span>940. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2273659">10.2307/2273659</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2273659">2273659</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:45660117">45660117</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Symbolic+Logic&rft.atitle=A+filter+lambda+model+and+the+completeness+of+type+assignment&rft.volume=48&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E931-%3C%2Fspan%3E940&rft.date=1983&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A45660117%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2273659%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2273659&rft.aulast=Barendregt&rft.aufirst=Henk&rft.au=Coppo%2C+Mario&rft.au=Dezani-Ciancaglini%2C+Mariangiola&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntersection+type" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPalsberg2012" class="citation book cs1">Palsberg, Jens (2012). "Overloading is NP-Complete". <i>Logic and Program Semantics</i>. Lecture Notes in Computer Science. Vol. 7230. pp. <span class="nowrap">204–</span>218. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-29485-3_13">10.1007/978-3-642-29485-3_13</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-29484-6" title="Special:BookSources/978-3-642-29484-6"><bdi>978-3-642-29484-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Overloading+is+NP-Complete&rft.btitle=Logic+and+Program+Semantics&rft.series=Lecture+Notes+in+Computer+Science&rft.pages=%3Cspan+class%3D%22nowrap%22%3E204-%3C%2Fspan%3E218&rft.date=2012&rft_id=info%3Adoi%2F10.1007%2F978-3-642-29485-3_13&rft.isbn=978-3-642-29484-6&rft.aulast=Palsberg&rft.aufirst=Jens&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntersection+type" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHenk_BarendregtWil_DekkersRichard_Statman2013" class="citation book cs1">Henk Barendregt; Wil Dekkers; Richard Statman (20 June 2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=2UVasvrhXl8C&pg=PR1"><i>Lambda Calculus with Types</i></a>. Cambridge University Press. pp. 1–. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-76614-2" title="Special:BookSources/978-0-521-76614-2"><bdi>978-0-521-76614-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lambda+Calculus+with+Types&rft.pages=1-&rft.pub=Cambridge+University+Press&rft.date=2013-06-20&rft.isbn=978-0-521-76614-2&rft.au=Henk+Barendregt&rft.au=Wil+Dekkers&rft.au=Richard+Statman&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D2UVasvrhXl8C%26pg%3DPR1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntersection+type" class="Z3988"></span></span> </li> <li id="cite_note-G96-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-G96_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGhilezan1996" class="citation journal cs1">Ghilezan, Silvia (1996). <a rel="nofollow" class="external text" href="https://doi.org/10.1305%2Fndjfl%2F1040067315">"Strong normalization and typability with intersection types"</a>. <i>Notre Dame Journal of Formal Logic</i>. <b>37</b> (1): <span class="nowrap">44–</span>52. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1305%2Fndjfl%2F1040067315">10.1305/ndjfl/1040067315</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Notre+Dame+Journal+of+Formal+Logic&rft.atitle=Strong+normalization+and+typability+with+intersection+types&rft.volume=37&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E44-%3C%2Fspan%3E52&rft.date=1996&rft_id=info%3Adoi%2F10.1305%2Fndjfl%2F1040067315&rft.aulast=Ghilezan&rft.aufirst=Silvia&rft_id=https%3A%2F%2Fdoi.org%2F10.1305%252Fndjfl%252F1040067315&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntersection+type" class="Z3988"></span></span> </li> <li id="cite_note-TS-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-TS_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-TS_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.typescriptlang.org/docs/handbook/advanced-types.html#intersection-types">"Intersection Types in TypeScript"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-08-01</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Intersection+Types+in+TypeScript&rft_id=https%3A%2F%2Fwww.typescriptlang.org%2Fdocs%2Fhandbook%2Fadvanced-types.html%23intersection-types&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntersection+type" class="Z3988"></span></span> </li> <li id="cite_note-K03-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-K03_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-K03_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKopylov2003" class="citation conference cs1">Kopylov, Alexei (2003). "Dependent intersection: A new way of defining records in type theory". <i>18th IEEE Symposium on Logic in Computer Science</i>. LICS 2003. IEEE Computer Society. pp. <span class="nowrap">86–</span>95. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.89.4223">10.1.1.89.4223</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FLICS.2003.1210048">10.1109/LICS.2003.1210048</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=Dependent+intersection%3A+A+new+way+of+defining+records+in+type+theory&rft.btitle=18th+IEEE+Symposium+on+Logic+in+Computer+Science&rft.pages=%3Cspan+class%3D%22nowrap%22%3E86-%3C%2Fspan%3E95&rft.pub=IEEE+Computer+Society&rft.date=2003&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.89.4223%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1109%2FLICS.2003.1210048&rft.aulast=Kopylov&rft.aufirst=Alexei&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntersection+type" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.scala-lang.org/files/archive/spec/2.12/04-basic-declarations-and-definitions.html#type-declarations-and-type-aliases">"Type declarations in Scala"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-08-15</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Type+declarations+in+Scala&rft_id=https%3A%2F%2Fwww.scala-lang.org%2Ffiles%2Farchive%2Fspec%2F2.12%2F04-basic-declarations-and-definitions.html%23type-declarations-and-type-aliases&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntersection+type" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAminGrütterOderskyRompf2016" class="citation book cs1">Amin, Nada; Grütter, Samuel; Odersky, Martin; Rompf, Tiark; Stucki, Sandro (2016). "The Essence of Dependent Object Types". <a rel="nofollow" class="external text" href="https://infoscience.epfl.ch/record/215280/files/paper_1.pdf"><i>A List of Successes That Can Change the World</i></a> <span class="cs1-format">(PDF)</span>. Lecture Notes in Computer Science. Vol. 9600. Springer. pp. <span class="nowrap">249–</span>272. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-30936-1_14">10.1007/978-3-319-30936-1_14</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-30935-4" title="Special:BookSources/978-3-319-30935-4"><bdi>978-3-319-30935-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Essence+of+Dependent+Object+Types&rft.btitle=A+List+of+Successes+That+Can+Change+the+World&rft.series=Lecture+Notes+in+Computer+Science&rft.pages=%3Cspan+class%3D%22nowrap%22%3E249-%3C%2Fspan%3E272&rft.pub=Springer&rft.date=2016&rft_id=info%3Adoi%2F10.1007%2F978-3-319-30936-1_14&rft.isbn=978-3-319-30935-4&rft.aulast=Amin&rft.aufirst=Nada&rft.au=Gr%C3%BCtter%2C+Samuel&rft.au=Odersky%2C+Martin&rft.au=Rompf%2C+Tiark&rft.au=Stucki%2C+Sandro&rft_id=https%3A%2F%2Finfoscience.epfl.ch%2Frecord%2F215280%2Ffiles%2Fpaper_1.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntersection+type" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://github.com/milessabin/shapeless/blob/master/core/src/main/scala/shapeless/singletons.scala">"Singletons in the Scala shapeless library"</a>. <i><a href="/wiki/GitHub" title="GitHub">GitHub</a></i><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-08-15</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=GitHub&rft.atitle=Singletons+in+the+Scala+shapeless+library&rft_id=https%3A%2F%2Fgithub.com%2Fmilessabin%2Fshapeless%2Fblob%2Fmaster%2Fcore%2Fsrc%2Fmain%2Fscala%2Fshapeless%2Fsingletons.scala&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntersection+type" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPollack2000" class="citation conference cs1">Pollack, Robert (2000). "Dependently typed records for representing mathematical structure". <i>Theorem Proving in Higher Order Logics, 13th International Conference</i>. TPHOLs 2000. Springer. pp. <span class="nowrap">462–</span>479. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-44659-1_29">10.1007/3-540-44659-1_29</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=Dependently+typed+records+for+representing+mathematical+structure&rft.btitle=Theorem+Proving+in+Higher+Order+Logics%2C+13th+International+Conference&rft.pages=%3Cspan+class%3D%22nowrap%22%3E462-%3C%2Fspan%3E479&rft.pub=Springer&rft.date=2000&rft_id=info%3Adoi%2F10.1007%2F3-540-44659-1_29&rft.aulast=Pollack&rft.aufirst=Robert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntersection+type" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFStump2018" class="citation journal cs1">Stump, Aaron (2018). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.apal.2018.03.002">"From realizability to induction via dependent intersection"</a>. <i>Annals of Pure and Applied Logic</i>. <b>169</b> (7): <span class="nowrap">637–</span>655. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.apal.2018.03.002">10.1016/j.apal.2018.03.002</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Pure+and+Applied+Logic&rft.atitle=From+realizability+to+induction+via+dependent+intersection&rft.volume=169&rft.issue=7&rft.pages=%3Cspan+class%3D%22nowrap%22%3E637-%3C%2Fspan%3E655&rft.date=2018&rft_id=info%3Adoi%2F10.1016%2Fj.apal.2018.03.002&rft.aulast=Stump&rft.aufirst=Aaron&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252Fj.apal.2018.03.002&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntersection+type" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://csharp.net/">"C# Guide"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-08-08</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=C%23+Guide&rft_id=http%3A%2F%2Fcsharp.net%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntersection+type" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://github.com/dotnet/csharplang/issues/399">"Discussion: Union and Intersection types in C Sharp"</a>. <i><a href="/wiki/GitHub" title="GitHub">GitHub</a></i><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-08-08</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=GitHub&rft.atitle=Discussion%3A+Union+and+Intersection+types+in+C+Sharp&rft_id=https%3A%2F%2Fgithub.com%2Fdotnet%2Fcsharplang%2Fissues%2F399&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntersection+type" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://projects.eclipse.org/projects/technology.ceylon/">"Eclipse Ceylon™"</a>. 19 July 2017<span class="reference-accessdate">. Retrieved <span class="nowrap">2023-08-16</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Eclipse+Ceylon%E2%84%A2&rft.date=2017-07-19&rft_id=https%3A%2F%2Fprojects.eclipse.org%2Fprojects%2Ftechnology.ceylon%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntersection+type" class="Z3988"></span></span> </li> <li id="cite_note-Cey-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-Cey_15-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ceylon-lang.org/documentation/tour/types/#intersection_types">"Intersection Types in Ceylon"</a>. 19 July 2017<span class="reference-accessdate">. Retrieved <span class="nowrap">2019-08-08</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Intersection+Types+in+Ceylon&rft.date=2017-07-19&rft_id=https%3A%2F%2Fceylon-lang.org%2Fdocumentation%2Ftour%2Ftypes%2F%23intersection_types&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntersection+type" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://fsharp.org/">"F# Software Foundation"</a><span class="reference-accessdate">. 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Retrieved <span class="nowrap">2019-08-08</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=GitHub&rft.atitle=Add+Intersection+Types+to+F+Sharp&rft_id=https%3A%2F%2Fgithub.com%2Ffsharp%2Ffslang-suggestions%2Fissues%2F600&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntersection+type" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20220408204727/https://flow.org/en/">"Flow: A Static Type Checker for JavaScript"</a>. Archived from <a rel="nofollow" class="external text" href="https://flow.org/en/">the original</a> on 2022-04-08<span class="reference-accessdate">. Retrieved <span class="nowrap">2019-08-08</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Flow%3A+A+Static+Type+Checker+for+JavaScript&rft_id=https%3A%2F%2Fflow.org%2Fen%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntersection+type" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://flow.org/en/docs/types/intersections/#intersection-type-syntax">"Intersection Type Syntax in Flow"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-08-08</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Intersection+Type+Syntax+in+Flow&rft_id=https%3A%2F%2Fflow.org%2Fen%2Fdocs%2Ftypes%2Fintersections%2F%23intersection-type-syntax&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntersection+type" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text">Reynolds, J. C. (1988). Preliminary design of the programming language Forsythe.</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.oracle.com/java/">"Java Software"</a><span class="reference-accessdate">. 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href="/wiki/Composite_data_type" title="Composite data type">Composite</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algebraic_data_type" title="Algebraic data type">Algebraic data type</a> <ul><li><a href="/wiki/Generalized_algebraic_data_type" title="Generalized algebraic data type">generalized</a></li></ul></li> <li><a href="/wiki/Array_data_type" class="mw-redirect" title="Array data type">Array</a></li> <li><a href="/wiki/Associative_array" title="Associative array">Associative array</a></li> <li><a href="/wiki/Class_(computer_programming)" title="Class (computer programming)">Class</a></li> <li><a href="/wiki/Dependent_type" title="Dependent type">Dependent</a></li> <li><a href="/wiki/Intuitionistic_type_theory#Equality_type" title="Intuitionistic type theory">Equality</a></li> <li><a href="/wiki/Inductive_type" title="Inductive type">Inductive</a></li> <li><a class="mw-selflink selflink">Intersection</a></li> <li><a href="/wiki/List_(abstract_data_type)" title="List (abstract data type)">List</a></li> <li><a href="/wiki/Object_(computer_science)" title="Object (computer science)">Object</a> <ul><li><a href="/wiki/Metaobject" title="Metaobject">metaobject</a></li></ul></li> <li><a href="/wiki/Option_type" title="Option type">Option type</a></li> <li><a href="/wiki/Product_type" title="Product type">Product</a></li> <li><a href="/wiki/Record_(computer_science)" title="Record (computer science)">Record or Struct</a></li> <li><a href="/wiki/Refinement_type" title="Refinement type">Refinement</a></li> <li><a href="/wiki/Set_(abstract_data_type)" title="Set (abstract data type)">Set</a></li> <li><a href="/wiki/Union_type" title="Union type">Union</a> <ul><li><a href="/wiki/Tagged_union" title="Tagged union">tagged</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boolean_data_type" title="Boolean data type">Boolean</a></li> <li><a href="/wiki/Bottom_type" title="Bottom type">Bottom type</a></li> <li><a href="/wiki/Container_(abstract_data_type)" title="Container (abstract data type)">Collection</a></li> <li><a href="/wiki/Enumerated_type" title="Enumerated type">Enumerated type</a></li> <li><a href="/wiki/Exception_handling" title="Exception handling">Exception</a></li> <li><a href="/wiki/Function_type" title="Function type">Function type</a></li> <li><a href="/wiki/Opaque_data_type" title="Opaque data type">Opaque data type</a></li> <li><a href="/wiki/Recursive_data_type" title="Recursive data type">Recursive data type</a></li> <li><a href="/wiki/Semaphore_(programming)" title="Semaphore (programming)">Semaphore</a></li> <li><a href="/wiki/Stream_(computing)" title="Stream (computing)">Stream</a></li> <li><a href="/wiki/Strongly_typed_identifier" title="Strongly typed identifier">Strongly typed identifier</a></li> <li><a href="/wiki/Top_type" title="Top type">Top type</a></li> <li><a href="/wiki/Type_class" title="Type class">Type class</a></li> <li><a href="/wiki/Empty_type" title="Empty type">Empty type</a></li> <li><a href="/wiki/Unit_type" title="Unit type">Unit type</a></li> <li><a href="/wiki/Void_type" title="Void type">Void</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related<br />topics</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_data_type" title="Abstract data type">Abstract data type</a></li> <li><a href="/wiki/Boxing_(computer_science)" class="mw-redirect" title="Boxing (computer science)">Boxing</a></li> <li><a href="/wiki/Data_structure" title="Data structure">Data structure</a></li> <li><a href="/wiki/Generic_programming" title="Generic programming">Generic</a></li> <li><a href="/wiki/Kind_(type_theory)" title="Kind (type theory)">Kind</a> <ul><li><a href="/wiki/Metaclass" title="Metaclass">metaclass</a></li></ul></li> <li><a href="/wiki/Parametric_polymorphism" title="Parametric polymorphism">Parametric polymorphism</a></li> <li><a href="/wiki/Primitive_data_type" title="Primitive data type">Primitive data type</a></li> <li><a href="/wiki/Interface_(object-oriented_programming)" title="Interface (object-oriented programming)">Interface</a></li> <li><a href="/wiki/Subtyping" title="Subtyping">Subtyping</a></li> <li><a href="/wiki/Type_constructor" title="Type constructor">Type constructor</a></li> <li><a href="/wiki/Type_conversion" title="Type conversion">Type conversion</a></li> <li><a href="/wiki/Type_system" title="Type system">Type system</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li> <li><a href="/wiki/Variable_(computer_science)" title="Variable (computer science)">Variable</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Intersection_type&oldid=1259099901">https://en.wikipedia.org/w/index.php?title=Intersection_type&oldid=1259099901</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Type_theory" title="Category:Type theory">Type theory</a></li><li><a href="/wiki/Category:Type_systems" title="Category:Type systems">Type systems</a></li><li><a href="/wiki/Category:Data_types" title="Category:Data types">Data types</a></li><li><a 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