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id="toc-Rank-1_(predicative)_polymorphism" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rank-1_(predicative)_polymorphism"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Rank-1 (predicative) polymorphism</span> </div> </a> <ul id="toc-Rank-1_(predicative)_polymorphism-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Higher-rank_polymorphism" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Higher-rank_polymorphism"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Higher-rank polymorphism</span> </div> </a> <ul id="toc-Higher-rank_polymorphism-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Impredicative_polymorphism" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Impredicative_polymorphism"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Impredicative polymorphism</span> </div> </a> <ul id="toc-Impredicative_polymorphism-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Bounded_parametric_polymorphism" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bounded_parametric_polymorphism"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Bounded parametric polymorphism</span> </div> </a> <ul id="toc-Bounded_parametric_polymorphism-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div 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.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="width:auto"><tbody><tr><th class="sidebar-title"><a href="/wiki/Polymorphism_(computer_science)" title="Polymorphism (computer science)">Polymorphism</a></th></tr><tr><th class="sidebar-heading" style="background-color:#ddf;font-size:110%"> <a href="/wiki/Ad_hoc_polymorphism" title="Ad hoc polymorphism">Ad hoc polymorphism</a></th></tr><tr><td class="sidebar-content plainlist"> <ul><li><a href="/wiki/Function_overloading" title="Function overloading">Function overloading</a></li> <li><a href="/wiki/Operator_overloading" title="Operator overloading">Operator overloading</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background-color:#ddf;font-size:110%"> <a class="mw-selflink selflink">Parametric polymorphism</a></th></tr><tr><td class="sidebar-content plainlist"> <ul><li><a href="/wiki/Generic_function" title="Generic function">Generic function</a></li> <li><a href="/wiki/Generic_programming" title="Generic programming">Generic programming</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background-color:#ddf;font-size:110%"> <a href="/wiki/Subtyping" title="Subtyping">Subtyping</a></th></tr><tr><td class="sidebar-content plainlist"> <ul><li><a href="/wiki/Virtual_function" title="Virtual function">Virtual function</a></li> <li><a href="/wiki/Dynamic_dispatch" title="Dynamic dispatch">Single and dynamic dispatch</a></li> <li><a href="/wiki/Double_dispatch" title="Double dispatch">Double dispatch</a></li> <li><a href="/wiki/Multiple_dispatch" title="Multiple dispatch">Multiple dispatch</a></li> <li><a href="/wiki/Predicate_dispatch" title="Predicate dispatch">Predicate dispatch</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:Polymorphism" title="Template:Polymorphism"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Polymorphism" title="Template talk:Polymorphism"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Polymorphism" title="Special:EditPage/Template:Polymorphism"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Programming_language" title="Programming language">programming languages</a> and <a href="/wiki/Type_theory" title="Type theory">type theory</a>, <b>parametric polymorphism</b> allows a single piece of code to be given a "generic" type, using variables in place of actual types, and then instantiated with particular types as needed.<sup id="cite_ref-TAPL_1-0" class="reference"><a href="#cite_note-TAPL-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 340">: 340 </span></sup> Parametrically polymorphic <a href="/wiki/Function_(programming)" class="mw-redirect" title="Function (programming)">functions</a> and <a href="/wiki/Data_type" title="Data type">data types</a> are sometimes called <b>generic functions</b> and <b>generic datatypes</b>, respectively, and they form the basis of <a href="/wiki/Generic_programming" title="Generic programming">generic programming</a>. </p><p>Parametric polymorphism may be contrasted with <a href="/wiki/Ad_hoc_polymorphism" title="Ad hoc polymorphism">ad hoc polymorphism</a>. Parametrically polymorphic definitions are <i>uniform</i>: they behave identically regardless of the type they are instantiated at.<sup id="cite_ref-TAPL_1-1" class="reference"><a href="#cite_note-TAPL-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 340">: 340 </span></sup><sup id="cite_ref-Strachey_1967_2-0" class="reference"><a href="#cite_note-Strachey_1967-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 37">: 37 </span></sup> In contrast, ad hoc polymorphic definitions are given a distinct definition for each type. Thus, ad hoc polymorphism can generally only support a limited number of such distinct types, since a separate implementation has to be provided for each type. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Basic_definition">Basic definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parametric_polymorphism&action=edit&section=1" title="Edit section: Basic definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is possible to write functions that do not depend on the types of their arguments. For example, the <a href="/wiki/Identity_function" title="Identity function">identity function</a> <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 {\mathsf {id}}(x)=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">d</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {id}}(x)=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e1dd57515ef496726df89998875096b3b7d325e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.325ex; height:2.843ex;" alt="{\displaystyle {\mathsf {id}}(x)=x}" /></span> simply returns its argument unmodified. This naturally gives rise to a family of potential types, such as <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 {\mathsf {Int}}\to {\mathsf {Int}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Int}}\to {\mathsf {Int}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39b3a0de4886cdd9859e72cd3382c3209367abc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.991ex; height:2.176ex;" alt="{\displaystyle {\mathsf {Int}}\to {\mathsf {Int}}}" /></span>, <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 {\mathsf {Bool}}\to {\mathsf {Bool}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">l</mi> </mrow> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">l</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Bool}}\to {\mathsf {Bool}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/278e85a130ed4ab4bcd11199557ab19a86cf3b97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.477ex; height:2.176ex;" alt="{\displaystyle {\mathsf {Bool}}\to {\mathsf {Bool}}}" /></span>, <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 {\mathsf {String}}\to {\mathsf {String}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">S</mi> <mi mathvariant="sans-serif">t</mi> <mi mathvariant="sans-serif">r</mi> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">g</mi> </mrow> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">S</mi> <mi mathvariant="sans-serif">t</mi> <mi mathvariant="sans-serif">r</mi> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {String}}\to {\mathsf {String}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ead8cac0ab896d3daa6987f5b48e4686a1e9fe47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.311ex; height:2.509ex;" alt="{\displaystyle {\mathsf {String}}\to {\mathsf {String}}}" /></span>, and so on. Parametric polymorphism allows <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 {\mathsf {id}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">d</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {id}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b0636dac4ec42814d578fb73a9e058400a3303e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.758ex; height:2.176ex;" alt="{\displaystyle {\mathsf {id}}}" /></span> to be given a single, <a href="/wiki/Most_general_unifier" class="mw-redirect" title="Most general unifier">most general</a> type by introducing a <a href="/wiki/Universal_quantification" title="Universal quantification">universally quantified</a> <a href="/wiki/Type_variable" title="Type variable">type variable</a>: </p> <dl><dd><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 {\mathsf {id}}:\forall \alpha .\alpha \to \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">d</mi> </mrow> </mrow> <mo>:</mo> <mi mathvariant="normal">∀</mi> <mi>α</mi> <mo>.</mo> <mi>α</mi> <mo stretchy="false">→</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {id}}:\forall \alpha .\alpha \to \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24075d02984405f4b0c329485c10486e81b9ba55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.099ex; height:2.176ex;" alt="{\displaystyle {\mathsf {id}}:\forall \alpha .\alpha \to \alpha }" /></span></dd></dl> <p>The polymorphic definition can then be <i>instantiated</i> by substituting any concrete type for <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></span>, yielding the full family of potential types.<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> </p><p>The identity function is a particularly extreme example, but many other functions also benefit from parametric polymorphism. For example, an <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 {\mathsf {append}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">p</mi> <mi mathvariant="sans-serif">p</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">d</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {append}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/281f8260ecacb65d954b5d7fe00503b043c28ca5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.958ex; height:2.509ex;" alt="{\displaystyle {\mathsf {append}}}" /></span> function that <a href="/wiki/Append" title="Append">concatenates</a> two <a href="/wiki/Linked_list" title="Linked list">lists</a> does not inspect the elements of the list, only the list structure itself. Therefore, <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 {\mathsf {append}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">p</mi> <mi mathvariant="sans-serif">p</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">d</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {append}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/281f8260ecacb65d954b5d7fe00503b043c28ca5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.958ex; height:2.509ex;" alt="{\displaystyle {\mathsf {append}}}" /></span> can be given a similar family of types, such as <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 [{\mathsf {Int}}]\times [{\mathsf {Int}}]\to [{\mathsf {Int}}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> <mo stretchy="false">]</mo> <mo>×</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [{\mathsf {Int}}]\times [{\mathsf {Int}}]\to [{\mathsf {Int}}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/772a5eb1e5ca02d1e18d3204673cd34422ecb34b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.401ex; height:2.843ex;" alt="{\displaystyle [{\mathsf {Int}}]\times [{\mathsf {Int}}]\to [{\mathsf {Int}}]}" /></span>, <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 [{\mathsf {Bool}}]\times [{\mathsf {Bool}}]\to [{\mathsf {Bool}}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">l</mi> </mrow> </mrow> <mo stretchy="false">]</mo> <mo>×</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">l</mi> </mrow> </mrow> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">l</mi> </mrow> </mrow> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [{\mathsf {Bool}}]\times [{\mathsf {Bool}}]\to [{\mathsf {Bool}}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f168b889085a07ae3b827b1ec98c0ce36a9d2145" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.63ex; height:2.843ex;" alt="{\displaystyle [{\mathsf {Bool}}]\times [{\mathsf {Bool}}]\to [{\mathsf {Bool}}]}" /></span>, and so on, 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 [T]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>T</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [T]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d2c8afd013f811776289b0d3f804a430888eccc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.93ex; height:2.843ex;" alt="{\displaystyle [T]}" /></span> denotes a list of elements 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 T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" /></span>. The most general type is therefore </p> <dl><dd><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 {\mathsf {append}}:\forall \alpha .[\alpha ]\times [\alpha ]\to [\alpha ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">p</mi> <mi mathvariant="sans-serif">p</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">d</mi> </mrow> </mrow> <mo>:</mo> <mi mathvariant="normal">∀</mi> <mi>α</mi> <mo>.</mo> <mo stretchy="false">[</mo> <mi>α</mi> <mo stretchy="false">]</mo> <mo>×</mo> <mo stretchy="false">[</mo> <mi>α</mi> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mi>α</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {append}}:\forall \alpha .[\alpha ]\times [\alpha ]\to [\alpha ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c312a5d6be210f84c1d5f883ec8573fa23d8474" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.508ex; height:2.843ex;" alt="{\displaystyle {\mathsf {append}}:\forall \alpha .[\alpha ]\times [\alpha ]\to [\alpha ]}" /></span></dd></dl> <p>which can be instantiated to any type in the family. </p><p>Parametrically polymorphic functions like <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 {\mathsf {id}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">d</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {id}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b0636dac4ec42814d578fb73a9e058400a3303e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.758ex; height:2.176ex;" alt="{\displaystyle {\mathsf {id}}}" /></span> and <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 {\mathsf {append}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">p</mi> <mi mathvariant="sans-serif">p</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">d</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {append}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/281f8260ecacb65d954b5d7fe00503b043c28ca5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.958ex; height:2.509ex;" alt="{\displaystyle {\mathsf {append}}}" /></span> are said to be <i>parameterized over</i> an arbitrary 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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></span>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Both <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 {\mathsf {id}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">d</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {id}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b0636dac4ec42814d578fb73a9e058400a3303e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.758ex; height:2.176ex;" alt="{\displaystyle {\mathsf {id}}}" /></span> and <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 {\mathsf {append}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">p</mi> <mi mathvariant="sans-serif">p</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">d</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {append}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/281f8260ecacb65d954b5d7fe00503b043c28ca5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.958ex; height:2.509ex;" alt="{\displaystyle {\mathsf {append}}}" /></span> are parameterized over a single type, but functions may be parameterized over arbitrarily many types. For example, the <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 {\mathsf {fst}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">f</mi> <mi mathvariant="sans-serif">s</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {fst}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a018bef5bd554003ffa9714192b65b5694c6c08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.537ex; height:2.176ex;" alt="{\displaystyle {\mathsf {fst}}}" /></span> and <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 {\mathsf {snd}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">s</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">d</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {snd}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c50af1832807d770e32b5b352d1eaebafc462a8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.295ex; height:2.176ex;" alt="{\displaystyle {\mathsf {snd}}}" /></span> functions that return the first and second elements of a <a href="/wiki/Product_type" title="Product type">pair</a>, respectively, can be given the following types: </p> <dl><dd><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 {\begin{aligned}{\mathsf {fst}}&:\forall \alpha .\forall \beta .\alpha \times \beta \to \alpha \\{\mathsf {snd}}&:\forall \alpha .\forall \beta .\alpha \times \beta \to \beta \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">f</mi> <mi mathvariant="sans-serif">s</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>:</mo> <mi mathvariant="normal">∀</mi> <mi>α</mi> <mo>.</mo> <mi mathvariant="normal">∀</mi> <mi>β</mi> <mo>.</mo> <mi>α</mi> <mo>×</mo> <mi>β</mi> <mo stretchy="false">→</mo> <mi>α</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">s</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">d</mi> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>:</mo> <mi mathvariant="normal">∀</mi> <mi>α</mi> <mo>.</mo> <mi mathvariant="normal">∀</mi> <mi>β</mi> <mo>.</mo> <mi>α</mi> <mo>×</mo> <mi>β</mi> <mo stretchy="false">→</mo> <mi>β</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\mathsf {fst}}&:\forall \alpha .\forall \beta .\alpha \times \beta \to \alpha \\{\mathsf {snd}}&:\forall \alpha .\forall \beta .\alpha \times \beta \to \beta \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97803ad0e98a028e106d9aaeab6ddd2d23fcf160" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.217ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}{\mathsf {fst}}&:\forall \alpha .\forall \beta .\alpha \times \beta \to \alpha \\{\mathsf {snd}}&:\forall \alpha .\forall \beta .\alpha \times \beta \to \beta \end{aligned}}}" /></span></dd></dl> <p>In the expression <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 {\mathsf {fst}}((3,{\mathsf {true}}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">f</mi> <mi mathvariant="sans-serif">s</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">t</mi> <mi mathvariant="sans-serif">r</mi> <mi mathvariant="sans-serif">u</mi> <mi mathvariant="sans-serif">e</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {fst}}((3,{\mathsf {true}}))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e72b15ee64d9ef2767b85d4471c537fdb3bbd86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.222ex; height:2.843ex;" alt="{\displaystyle {\mathsf {fst}}((3,{\mathsf {true}}))}" /></span>, <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></span> is instantiated to <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 {\mathsf {Int}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Int}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6a1b1c2eb940eb7b2f17e2c3837ea7e45ecb16b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.688ex; height:2.176ex;" alt="{\displaystyle {\mathsf {Int}}}" /></span> and <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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" /></span> is instantiated to <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 {\mathsf {Bool}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">l</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Bool}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79ce10cc2b8497d68990945cd3b63c9909837ae7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.431ex; height:2.176ex;" alt="{\displaystyle {\mathsf {Bool}}}" /></span> in the call to <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 {\mathsf {fst}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">f</mi> <mi mathvariant="sans-serif">s</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {fst}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a018bef5bd554003ffa9714192b65b5694c6c08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.537ex; height:2.176ex;" alt="{\displaystyle {\mathsf {fst}}}" /></span>, so the type of the overall expression is <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 {\mathsf {Int}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Int}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6a1b1c2eb940eb7b2f17e2c3837ea7e45ecb16b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.688ex; height:2.176ex;" alt="{\displaystyle {\mathsf {Int}}}" /></span>. </p><p>The <a href="/wiki/Syntax_(programming_languages)" title="Syntax (programming languages)">syntax</a> used to introduce parametric polymorphism varies significantly between programming languages. For example, in some programming languages, such as <a href="/wiki/Haskell" title="Haskell">Haskell</a>, the <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 \forall \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bea60e28fc9299c5c885c473f025a858cb676a79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.78ex; height:2.176ex;" alt="{\displaystyle \forall \alpha }" /></span> <a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">quantifier</a> is implicit and may be omitted.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Other languages require types to be instantiated explicitly at some or all of a parametrically polymorphic function's <a href="/wiki/Call_site" title="Call site">call sites</a>. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parametric_polymorphism&action=edit&section=2" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Parametric polymorphism was first introduced to programming languages in <a href="/wiki/ML_programming_language" class="mw-redirect" title="ML programming language">ML</a> in 1975.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Today it exists in <a href="/wiki/Standard_ML" title="Standard ML">Standard ML</a>, <a href="/wiki/OCaml" title="OCaml">OCaml</a>, <a href="/wiki/F_Sharp_(programming_language)" title="F Sharp (programming language)">F#</a>, <a href="/wiki/Ada_(programming_language)" title="Ada (programming language)">Ada</a>, <a href="/wiki/Haskell_(programming_language)" class="mw-redirect" title="Haskell (programming language)">Haskell</a>, <a href="/wiki/Mercury_(programming_language)" title="Mercury (programming language)">Mercury</a>, <a href="/wiki/Visual_Prolog" title="Visual Prolog">Visual Prolog</a>, <a href="/wiki/Scala_(programming_language)" title="Scala (programming language)">Scala</a>, <a href="/wiki/Julia_(programming_language)" title="Julia (programming language)">Julia</a>, <a href="/wiki/Python_(programming_language)" title="Python (programming language)">Python</a>, <a href="/wiki/TypeScript" title="TypeScript">TypeScript</a>, <a href="/wiki/C%2B%2B" title="C++">C++</a> and others. <a href="/wiki/Java_(programming_language)" title="Java (programming language)">Java</a>, <a href="/wiki/C_Sharp_(programming_language)" title="C Sharp (programming language)">C#</a>, <a href="/wiki/Visual_Basic_.NET" class="mw-redirect" title="Visual Basic .NET">Visual Basic .NET</a> and <a href="/wiki/Object_Pascal" title="Object Pascal">Delphi</a> have each introduced "generics" for parametric polymorphism. Some implementations of type polymorphism are superficially similar to parametric polymorphism while also introducing ad hoc aspects. One example is <a href="/wiki/C%2B%2B" title="C++">C++</a> <a href="/wiki/Template_specialization" class="mw-redirect" title="Template specialization">template specialization</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Predicativity,_impredicativity,_and_higher-rank_polymorphism"><span id="Predicativity.2C_impredicativity.2C_and_higher-rank_polymorphism"></span>Predicativity, impredicativity, and higher-rank polymorphism</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parametric_polymorphism&action=edit&section=3" title="Edit section: Predicativity, impredicativity, and higher-rank polymorphism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Rank-1_(predicative)_polymorphism"><span id="Rank-1_.28predicative.29_polymorphism"></span>Rank-1 (predicative) polymorphism</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parametric_polymorphism&action=edit&section=4" title="Edit section: Rank-1 (predicative) polymorphism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed_section plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Parametric_polymorphism" title="Special:EditPage/Parametric polymorphism">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a> in this section. Unsourced material may be challenged and removed.</span> <span class="date-container"><i>(<span class="date">February 2019</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In a <i><a href="/wiki/Impredicativity" title="Impredicativity">predicative</a></i> type system (also known as a <i><a href="/wiki/Prenex" class="mw-redirect" title="Prenex">prenex</a> polymorphic</i> system), type variables may not be instantiated with polymorphic types.<sup id="cite_ref-TAPL_1-2" class="reference"><a href="#cite_note-TAPL-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Pages: 359–360">: 359–360 </span></sup> Predicative type theories include <a href="/wiki/Intuitionistic_type_theory" title="Intuitionistic type theory">Martin-Löf type theory</a> and <a href="/wiki/Nuprl" title="Nuprl">Nuprl</a>. This is very similar to what is called "ML-style" or "Let-polymorphism" (technically ML's Let-polymorphism has a few other syntactic restrictions). This restriction makes the distinction between polymorphic and non-polymorphic types very important; thus in predicative systems polymorphic types are sometimes referred to as <i>type schemas</i> to distinguish them from ordinary (monomorphic) types, which are sometimes called <i>monotypes</i>. </p><p>A consequence of predicativity is that all types can be written in a form that places all quantifiers at the outermost (prenex) position. For example, consider the <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 {\mathsf {append}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">p</mi> <mi mathvariant="sans-serif">p</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">d</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {append}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/281f8260ecacb65d954b5d7fe00503b043c28ca5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.958ex; height:2.509ex;" alt="{\displaystyle {\mathsf {append}}}" /></span> function described above, which has the following type: </p> <dl><dd><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 {\mathsf {append}}:\forall \alpha .[\alpha ]\times [\alpha ]\to [\alpha ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">p</mi> <mi mathvariant="sans-serif">p</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">d</mi> </mrow> </mrow> <mo>:</mo> <mi mathvariant="normal">∀</mi> <mi>α</mi> <mo>.</mo> <mo stretchy="false">[</mo> <mi>α</mi> <mo stretchy="false">]</mo> <mo>×</mo> <mo stretchy="false">[</mo> <mi>α</mi> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mi>α</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {append}}:\forall \alpha .[\alpha ]\times [\alpha ]\to [\alpha ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c312a5d6be210f84c1d5f883ec8573fa23d8474" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.508ex; height:2.843ex;" alt="{\displaystyle {\mathsf {append}}:\forall \alpha .[\alpha ]\times [\alpha ]\to [\alpha ]}" /></span></dd></dl> <p>In order to apply this function to a pair of lists, a concrete 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 T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" /></span> must be substituted for the 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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></span> such that the resulting function type is consistent with the types of the arguments. In an <i>impredicative</i> system, <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 T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" /></span> may be any type whatsoever, including a type that is itself polymorphic; thus <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 {\mathsf {append}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">p</mi> <mi mathvariant="sans-serif">p</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">d</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {append}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/281f8260ecacb65d954b5d7fe00503b043c28ca5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.958ex; height:2.509ex;" alt="{\displaystyle {\mathsf {append}}}" /></span> can be applied to pairs of lists with elements of any type—even to lists of polymorphic functions such as <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 {\mathsf {append}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">p</mi> <mi mathvariant="sans-serif">p</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">d</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {append}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/281f8260ecacb65d954b5d7fe00503b043c28ca5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.958ex; height:2.509ex;" alt="{\displaystyle {\mathsf {append}}}" /></span> itself. Polymorphism in the language ML is predicative.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (February 2019)">citation needed</span></a></i>]</sup> This is because predicativity, together with other restrictions, makes the <a href="/wiki/Type_system" title="Type system">type system</a> simple enough that full <a href="/wiki/Type_inference" title="Type inference">type inference</a> is always possible. </p><p>As a practical example, <a href="/wiki/OCaml" title="OCaml">OCaml</a> (a descendant or dialect of <a href="/wiki/ML_(programming_language)" title="ML (programming language)">ML</a>) performs type inference and supports impredicative polymorphism, but in some cases when impredicative polymorphism is used, the system's type inference is incomplete unless some explicit type annotations are provided by the programmer. </p> <div class="mw-heading mw-heading3"><h3 id="Higher-rank_polymorphism">Higher-rank polymorphism</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parametric_polymorphism&action=edit&section=5" title="Edit section: Higher-rank polymorphism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some type systems support an impredicative function type constructor even though other type constructors remain predicative. For example, 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 (\forall \alpha .\alpha \rightarrow \alpha )\rightarrow T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">∀</mi> <mi>α</mi> <mo>.</mo> <mi>α</mi> <mo stretchy="false">→</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\forall \alpha .\alpha \rightarrow \alpha )\rightarrow T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70351b3886a1b5c350206f274432926b681f9f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.463ex; height:2.843ex;" alt="{\displaystyle (\forall \alpha .\alpha \rightarrow \alpha )\rightarrow T}" /></span> is permitted in a system that supports higher-rank polymorphism, even though <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 [\forall \alpha .\alpha \rightarrow \alpha ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi mathvariant="normal">∀</mi> <mi>α</mi> <mo>.</mo> <mi>α</mi> <mo stretchy="false">→</mo> <mi>α</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [\forall \alpha .\alpha \rightarrow \alpha ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c90e9eda4c5e4b9196d2d3311ca6b8f7fcb5241" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.697ex; height:2.843ex;" alt="{\displaystyle [\forall \alpha .\alpha \rightarrow \alpha ]}" /></span> may not be.<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> </p><p>A type is said to be of rank <i>k</i> (for some fixed integer <i>k</i>) if no path from its root to a <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 \forall }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfc1a1a9c4c0f8d5df989c98aa2773ed657c5937" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \forall }" /></span> quantifier passes to the left of <i>k</i> or more arrows, when the type is drawn as a tree.<sup id="cite_ref-TAPL_1-3" class="reference"><a href="#cite_note-TAPL-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 359">: 359 </span></sup> A type system is said to support rank-<i>k</i> polymorphism if it admits types with rank less than or equal to <i>k</i>. For example, a type system that supports rank-2 polymorphism would allow <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 (\forall \alpha .\alpha \rightarrow \alpha )\rightarrow T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">∀</mi> <mi>α</mi> <mo>.</mo> <mi>α</mi> <mo stretchy="false">→</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\forall \alpha .\alpha \rightarrow \alpha )\rightarrow T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70351b3886a1b5c350206f274432926b681f9f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.463ex; height:2.843ex;" alt="{\displaystyle (\forall \alpha .\alpha \rightarrow \alpha )\rightarrow T}" /></span> but not <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 ((\forall \alpha .\alpha \rightarrow \alpha )\rightarrow T)\rightarrow T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∀</mi> <mi>α</mi> <mo>.</mo> <mi>α</mi> <mo stretchy="false">→</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((\forall \alpha .\alpha \rightarrow \alpha )\rightarrow T)\rightarrow T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da7d6756098831ffb772b89e9773b9b5f8670d50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.523ex; height:2.843ex;" alt="{\displaystyle ((\forall \alpha .\alpha \rightarrow \alpha )\rightarrow T)\rightarrow T}" /></span>. A type system that admits types of arbitrary rank is said to be "rank-<i>n</i> polymorphic". </p><p><a href="/wiki/Type_inference" title="Type inference">Type inference</a> for rank-2 polymorphism is decidable, but for rank-3 and above, it is not.<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><sup id="cite_ref-TAPL_1-4" class="reference"><a href="#cite_note-TAPL-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 359">: 359 </span></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Impredicative_polymorphism">Impredicative polymorphism</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parametric_polymorphism&action=edit&section=6" title="Edit section: Impredicative polymorphism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>Impredicative polymorphism</i> (also called <i>first-class polymorphism</i>) is the most powerful form of parametric polymorphism.<sup id="cite_ref-TAPL_1-5" class="reference"><a href="#cite_note-TAPL-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 340">: 340 </span></sup> In <a href="/wiki/Formal_logic" class="mw-redirect" title="Formal logic">formal logic</a>, a definition is said to be <a href="/wiki/Impredicativity" title="Impredicativity">impredicative</a> if it is self-referential; in type theory, it refers to the ability for a type to be in the domain of a quantifier it contains. This allows the instantiation of any type variable with any type, including polymorphic types. An example of a system supporting full impredicativity is <a href="/wiki/System_F" title="System F">System F</a>, which allows instantiating <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 \forall \alpha .\alpha \to \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>α</mi> <mo>.</mo> <mi>α</mi> <mo stretchy="false">→</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \alpha .\alpha \to \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d4efbef53da2a6436dc0e5962f3b12fe0de8f22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.403ex; height:2.176ex;" alt="{\displaystyle \forall \alpha .\alpha \to \alpha }" /></span> at any type, including itself. </p><p>In <a href="/wiki/Type_theory" title="Type theory">type theory</a>, the most frequently studied impredicative <a href="/wiki/Typed_lambda_calculus" title="Typed lambda calculus">typed λ-calculi</a> are based on those of the <a href="/wiki/Lambda_cube" title="Lambda cube">lambda cube</a>, especially System F. </p> <div class="mw-heading mw-heading2"><h2 id="Bounded_parametric_polymorphism">Bounded parametric polymorphism</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parametric_polymorphism&action=edit&section=7" title="Edit section: Bounded parametric polymorphism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Bounded_quantification" title="Bounded quantification">Bounded quantification</a></div> <p>In 1985, <a href="/wiki/Luca_Cardelli" title="Luca Cardelli">Luca Cardelli</a> and <a href="/wiki/Peter_Wegner_(computer_scientist)" title="Peter Wegner (computer scientist)">Peter Wegner</a> recognized the advantages of allowing <i>bounds</i> on the type parameters.<sup id="cite_ref-FOOTNOTECardelliWegner1985_9-0" class="reference"><a href="#cite_note-FOOTNOTECardelliWegner1985-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Many operations require some knowledge of the data types, but can otherwise work parametrically. For example, to check whether an item is included in a list, we need to compare the items for equality. In <a href="/wiki/Standard_ML" title="Standard ML">Standard ML</a>, type parameters of the form <i>’’a</i> are restricted so that the equality operation is available, thus the function would have the type <i>’’a</i> × <i>’’a</i> list → bool and <i>’’a</i> can only be a type with defined equality. In <a href="/wiki/Haskell_(programming_language)" class="mw-redirect" title="Haskell (programming language)">Haskell</a>, bounding is achieved by requiring types to belong to a <a href="/wiki/Type_class" title="Type class">type class</a>; thus the same function 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 \mathrm {Eq} \,\alpha \,\Rightarrow \alpha \,\rightarrow \left[\alpha \right]\rightarrow \mathrm {Bool} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">q</mi> </mrow> <mspace width="thinmathspace"></mspace> <mi>α</mi> <mspace width="thinmathspace"></mspace> <mo stretchy="false">⇒</mo> <mi>α</mi> <mspace width="thinmathspace"></mspace> <mo stretchy="false">→</mo> <mrow> <mo>[</mo> <mi>α</mi> <mo>]</mo> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">l</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \mathrm {Eq} \,\alpha \,\Rightarrow \alpha \,\rightarrow \left[\alpha \right]\rightarrow \mathrm {Bool} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c71d9c77a957c312767e7af5eccb082c66a3892" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.204ex; height:2.843ex;" alt="{\textstyle \mathrm {Eq} \,\alpha \,\Rightarrow \alpha \,\rightarrow \left[\alpha \right]\rightarrow \mathrm {Bool} }" /></span> in Haskell. In most object-oriented programming languages that support parametric polymorphism, parameters can be constrained to be <a href="/wiki/Subtyping" title="Subtyping">subtypes</a> of a given type (see the articles <a href="/wiki/Subtype_polymorphism" class="mw-redirect" title="Subtype polymorphism">Subtype polymorphism</a> and <a href="/wiki/Generic_programming" title="Generic programming">Generic programming</a>). </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parametric_polymorphism&action=edit&section=8" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Parametricity" title="Parametricity">Parametricity</a></li> <li><a href="/wiki/Polymorphic_recursion" title="Polymorphic recursion">Polymorphic recursion</a></li> <li><a href="/wiki/Type_class#Higher-kinded_polymorphism" title="Type class">Type class#Higher-kinded polymorphism</a></li> <li><a href="/wiki/Trait_(computer_programming)" title="Trait (computer programming)">Trait (computer programming)</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parametric_polymorphism&action=edit&section=9" title="Edit section: Notes"><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 reflist-columns references-column-width reflist-columns-2"> <ol class="references"> <li id="cite_note-TAPL-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-TAPL_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-TAPL_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-TAPL_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-TAPL_1-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-TAPL_1-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-TAPL_1-5"><sup><i><b>f</b></i></sup></a></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="CITEREFBenjamin_C._Pierce2002" class="citation book cs1"><a href="/wiki/Benjamin_C._Pierce" title="Benjamin C. Pierce">Benjamin C. Pierce</a> (2002). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ti6zoAC9Ph8C"><i>Types and Programming Languages</i></a>. MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-16209-8" title="Special:BookSources/978-0-262-16209-8"><bdi>978-0-262-16209-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Types+and+Programming+Languages&rft.pub=MIT+Press&rft.date=2002&rft.isbn=978-0-262-16209-8&rft.au=Benjamin+C.+Pierce&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dti6zoAC9Ph8C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParametric+polymorphism" class="Z3988"></span></span> </li> <li id="cite_note-Strachey_1967-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Strachey_1967_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFStrachey1967" class="citation cs2"><a href="/wiki/Christopher_Strachey" title="Christopher Strachey">Strachey, Christopher</a> (1967), <a href="/wiki/Fundamental_Concepts_in_Programming_Languages" title="Fundamental Concepts in Programming Languages"><i>Fundamental Concepts in Programming Languages</i></a> (Lecture notes), Copenhagen: International Summer School in Computer Programming</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamental+Concepts+in+Programming+Languages&rft.place=Copenhagen&rft.pub=International+Summer+School+in+Computer+Programming&rft.date=1967&rft.aulast=Strachey&rft.aufirst=Christopher&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParametric+polymorphism" class="Z3988"></span>. 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Retrieved <span class="nowrap">1 October</span> 2022</span>. <q>With one exception (that of the distinguished type variable in a class declaration (Section 4.3.1)), the type variables in a Haskell type expression are all assumed to be universally quantified; there is no explicit syntax for universal quantification.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.haskell.org&rft.atitle=Haskell+2010+Language+Report+%C2%A7+4.1.2+Syntax+of+Types&rft_id=https%3A%2F%2Fwww.haskell.org%2Fonlinereport%2Fhaskell2010%2Fhaskellch4.html%23x10-650004.1.2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParametric+polymorphism" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="/wiki/Robin_Milner" title="Robin Milner">Milner, R.</a>, Morris, L., Newey, M. 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"Principality and decidable type inference for finite-rank intersection types". <i>Proceedings of the 26th ACM SIGPLAN-SIGACT <a href="/wiki/Symposium_on_Principles_of_Programming_Languages" title="Symposium on Principles of Programming Languages">Symposium on Principles of Programming Languages</a></i>. Association for Computing Machinery. pp. <span class="nowrap">161–</span>174. <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/10.1145%2F292540.292556">10.1145/292540.292556</a></span>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1581130953" title="Special:BookSources/1581130953"><bdi>1581130953</bdi></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:14183560">14183560</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Principality+and+decidable+type+inference+for+finite-rank+intersection+types&rft.btitle=Proceedings+of+the+26th+ACM+SIGPLAN-SIGACT+Symposium+on+Principles+of+Programming+Languages&rft.pages=%3Cspan+class%3D%22nowrap%22%3E161-%3C%2Fspan%3E174&rft.pub=Association+for+Computing+Machinery&rft.date=1999-01-01&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14183560%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1145%2F292540.292556&rft.isbn=1581130953&rft.aulast=Kfoury&rft.aufirst=A.+J.&rft.au=Wells%2C+J.+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParametric+polymorphism" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTECardelliWegner1985-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECardelliWegner1985_9-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCardelliWegner1985">Cardelli & Wegner 1985</a>.</span> </li> </ol></div> <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=Parametric_polymorphism&action=edit&section=10" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li class="mw-empty-elt"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHindley1969" class="citation cs2"><a href="/wiki/J._Roger_Hindley" title="J. 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Amsterdam. pp. <span class="nowrap">63–</span>92. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0049-237X%2808%2970843-7">10.1016/S0049-237X(08)70843-7</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780720422597" title="Special:BookSources/9780720422597"><bdi>9780720422597</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=Une+Extension+de+l%27Interpretation+de+G%C3%B6del+%C3%A0+l%27Analyse%2C+et+son+Application+%C3%A0+l%27%C3%89limination+des+Coupures+dans+l%27Analyse+et+la+Th%C3%A9orie+des+Types&rft.btitle=Proceedings+of+the+Second+Scandinavian+Logic+Symposium&rft.place=Amsterdam&rft.series=Studies+in+Logic+and+the+Foundations+of+Mathematics&rft.pages=%3Cspan+class%3D%22nowrap%22%3E63-%3C%2Fspan%3E92&rft.date=1971&rft_id=info%3Adoi%2F10.1016%2FS0049-237X%2808%2970843-7&rft.isbn=9780720422597&rft.aulast=Girard&rft.aufirst=Jean-Yves&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParametric+polymorphism" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGirard1972" class="citation cs2 cs1-prop-foreign-lang-source"><a href="/wiki/Jean-Yves_Girard" title="Jean-Yves Girard">Girard, Jean-Yves</a> (1972), <i>Interprétation fonctionnelle et élimination des coupures de l'arithmétique d'ordre supérieur</i> (Ph.D. thesis) (in French), Université Paris 7</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Interpr%C3%A9tation+fonctionnelle+et+%C3%A9limination+des+coupures+de+l%27arithm%C3%A9tique+d%27ordre+sup%C3%A9rieur&rft.pub=Universit%C3%A9+Paris+7&rft.date=1972&rft.aulast=Girard&rft.aufirst=Jean-Yves&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParametric+polymorphism" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFReynolds1974" class="citation cs2"><a href="/wiki/John_C._Reynolds" title="John C. Reynolds">Reynolds, John C.</a> (1974), <a rel="nofollow" class="external text" href="https://figshare.com/articles/journal_contribution/6611015">"Towards a Theory of Type Structure"</a>, <i>Colloque Sur la Programmation</i>, <a href="/wiki/Lecture_Notes_in_Computer_Science" title="Lecture Notes in Computer Science">Lecture Notes in Computer Science</a>, <b>19</b>, Paris: <span class="nowrap">408–</span>425, <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%2F3-540-06859-7_148">10.1007/3-540-06859-7_148</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-06859-4" title="Special:BookSources/978-3-540-06859-4"><bdi>978-3-540-06859-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Colloque+Sur+la+Programmation&rft.atitle=Towards+a+Theory+of+Type+Structure&rft.volume=19&rft.pages=%3Cspan+class%3D%22nowrap%22%3E408-%3C%2Fspan%3E425&rft.date=1974&rft_id=info%3Adoi%2F10.1007%2F3-540-06859-7_148&rft.isbn=978-3-540-06859-4&rft.aulast=Reynolds&rft.aufirst=John+C.&rft_id=https%3A%2F%2Ffigshare.com%2Farticles%2Fjournal_contribution%2F6611015&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParametric+polymorphism" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMilner1978" class="citation journal cs1"><a href="/wiki/Robin_Milner" title="Robin Milner">Milner, Robin</a> (1978). <a rel="nofollow" class="external text" href="https://www.pure.ed.ac.uk/ws/files/15143545/1_s2.0_0022000078900144_main.pdf">"A Theory of Type Polymorphism in Programming"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Journal_of_Computer_and_System_Sciences" title="Journal of Computer and System Sciences">Journal of Computer and System Sciences</a></i>. <b>17</b> (3): <span class="nowrap">348–</span>375. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0022-0000%2878%2990014-4">10.1016/0022-0000(78)90014-4</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:388583">388583</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+Computer+and+System+Sciences&rft.atitle=A+Theory+of+Type+Polymorphism+in+Programming&rft.volume=17&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E348-%3C%2Fspan%3E375&rft.date=1978&rft_id=info%3Adoi%2F10.1016%2F0022-0000%2878%2990014-4&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A388583%23id-name%3DS2CID&rft.aulast=Milner&rft.aufirst=Robin&rft_id=https%3A%2F%2Fwww.pure.ed.ac.uk%2Fws%2Ffiles%2F15143545%2F1_s2.0_0022000078900144_main.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParametric+polymorphism" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCardelliWegner1985" class="citation journal cs1"><a href="/wiki/Luca_Cardelli" title="Luca Cardelli">Cardelli, Luca</a>; <a href="/wiki/Peter_Wegner_(computer_scientist)" title="Peter Wegner (computer scientist)">Wegner, Peter</a> (December 1985). <a rel="nofollow" class="external text" href="http://lucacardelli.name/Papers/OnUnderstanding.A4.pdf">"On Understanding Types, Data Abstraction, and Polymorphism"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/ACM_Computing_Surveys" title="ACM Computing Surveys">ACM Computing Surveys</a></i>. <b>17</b> (4): <span class="nowrap">471–</span>523. <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=10.1.1.117.695">10.1.1.117.695</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/10.1145%2F6041.6042">10.1145/6041.6042</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0360-0300">0360-0300</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:2921816">2921816</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=ACM+Computing+Surveys&rft.atitle=On+Understanding+Types%2C+Data+Abstraction%2C+and+Polymorphism&rft.volume=17&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E471-%3C%2Fspan%3E523&rft.date=1985-12&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.117.695%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A2921816%23id-name%3DS2CID&rft.issn=0360-0300&rft_id=info%3Adoi%2F10.1145%2F6041.6042&rft.aulast=Cardelli&rft.aufirst=Luca&rft.au=Wegner%2C+Peter&rft_id=http%3A%2F%2Flucacardelli.name%2FPapers%2FOnUnderstanding.A4.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParametric+polymorphism" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPierce2002" class="citation book cs1"><a href="/wiki/Benjamin_C._Pierce" title="Benjamin C. Pierce">Pierce, Benjamin C.</a> (2002). <i>Types and Programming Languages</i>. MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-16209-8" title="Special:BookSources/978-0-262-16209-8"><bdi>978-0-262-16209-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Types+and+Programming+Languages&rft.pub=MIT+Press&rft.date=2002&rft.isbn=978-0-262-16209-8&rft.aulast=Pierce&rft.aufirst=Benjamin+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParametric+polymorphism" class="Z3988"></span></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output 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style="width:1%"><a href="/wiki/Units_of_information" title="Units of information">Uninterpreted</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/Bit" title="Bit">Bit</a></li> <li><a href="/wiki/Byte" title="Byte">Byte</a></li> <li><a href="/wiki/Ternary_numeral_system" title="Ternary numeral system">Trit</a></li> <li><a href="/wiki/Ternary_numeral_system#Tryte" title="Ternary numeral system">Tryte</a></li> <li><a href="/wiki/Word_(computer_architecture)" title="Word (computer architecture)">Word</a></li> <li><a href="/wiki/Bit_array" title="Bit array">Bit array</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Numeric</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/Arbitrary-precision_arithmetic" title="Arbitrary-precision arithmetic">Arbitrary-precision or bignum</a></li> <li><a href="/wiki/Complex_data_type" title="Complex data type">Complex</a></li> <li><a href="/wiki/Decimal_data_type" title="Decimal data type">Decimal</a></li> <li><a href="/wiki/Fixed-point_arithmetic" title="Fixed-point arithmetic">Fixed point</a></li> <li><a href="/wiki/Floating-point_arithmetic" title="Floating-point arithmetic">Floating point</a> <ul><li>Reduced precision <ul><li><a href="/wiki/Minifloat" title="Minifloat">Minifloat</a></li> <li><a href="/wiki/Half-precision_floating-point_format" title="Half-precision floating-point format">Half precision</a></li> <li><a href="/wiki/Bfloat16_floating-point_format" title="Bfloat16 floating-point format">bfloat16</a></li></ul></li> <li><a href="/wiki/Single-precision_floating-point_format" title="Single-precision floating-point format">Single precision</a></li> <li><a href="/wiki/Double-precision_floating-point_format" title="Double-precision floating-point format">Double precision</a></li> <li><a href="/wiki/Quadruple-precision_floating-point_format" title="Quadruple-precision floating-point format">Quadruple precision</a></li> <li><a href="/wiki/Octuple-precision_floating-point_format" title="Octuple-precision floating-point format">Octuple precision</a></li> <li><a href="/wiki/Extended_precision" title="Extended precision">Extended precision</a> <ul><li><a href="/wiki/Long_double" title="Long double">Long double</a></li></ul></li></ul></li> <li><a href="/wiki/Integer_(computer_science)" title="Integer (computer science)">Integer</a> <ul><li><a href="/wiki/Signedness" title="Signedness">signedness</a></li></ul></li> <li><a href="/wiki/Interval_arithmetic#Implementations" title="Interval arithmetic">Interval</a></li> <li><a href="/wiki/Rational_data_type" title="Rational data type">Rational</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Pointer_(computer_programming)" title="Pointer (computer programming)">Pointer</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/Memory_address" title="Memory address">Address</a> <ul><li><a href="/wiki/Physical_address" title="Physical address">physical</a></li> <li><a href="/wiki/Virtual_address_space" title="Virtual address space">virtual</a></li></ul></li> <li><a href="/wiki/Reference_(computer_science)" title="Reference (computer science)">Reference</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Plain_text" title="Plain text">Text</a></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/Character_(computing)" title="Character (computing)">Character</a></li> <li><a href="/wiki/String_(computer_science)" title="String (computer science)">String</a> <ul><li><a href="/wiki/Null-terminated_string" title="Null-terminated string">null-terminated</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a 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 href="/wiki/Intersection_type" title="Intersection type">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)" 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