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1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> dependent type theory </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/3311/#Item_11" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="type_theory">Type theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/natural+deduction">natural deduction</a></strong> <a class="existingWikiWord" href="/nlab/show/metalanguage">metalanguage</a>, <a class="existingWikiWord" href="/nlab/show/practical+foundations">practical foundations</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/judgement">judgement</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypothetical+judgement">hypothetical judgement</a>, <a class="existingWikiWord" href="/nlab/show/sequent">sequent</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/antecedents">antecedents</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊢</mo></mrow><annotation encoding="application/x-tex">\vdash</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/consequent">consequent</a>, <a class="existingWikiWord" href="/nlab/show/succedents">succedents</a></li> </ul> </li> </ul> <ol> <li><a class="existingWikiWord" href="/nlab/show/type+formation+rule">type formation rule</a></li> <li><a class="existingWikiWord" href="/nlab/show/term+introduction+rule">term introduction rule</a></li> <li><a class="existingWikiWord" href="/nlab/show/term+elimination+rule">term elimination rule</a></li> <li><a class="existingWikiWord" href="/nlab/show/computation+rule">computation rule</a></li> </ol> <p><strong><a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a></strong> (<a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent</a>, <a class="existingWikiWord" href="/nlab/show/intensional+type+theory">intensional</a>, <a class="existingWikiWord" href="/nlab/show/observational+type+theory">observational type theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>)</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/calculus+of+constructions">calculus of constructions</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/syntax">syntax</a></strong> <a class="existingWikiWord" href="/nlab/show/object+language">object language</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/theory">theory</a>, <a class="existingWikiWord" href="/nlab/show/axiom">axiom</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proposition">proposition</a>/<a class="existingWikiWord" href="/nlab/show/type">type</a> (<a class="existingWikiWord" href="/nlab/show/propositions+as+types">propositions as types</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/definition">definition</a>/<a class="existingWikiWord" href="/nlab/show/proof">proof</a>/<a class="existingWikiWord" href="/nlab/show/program">program</a> (<a class="existingWikiWord" href="/nlab/show/proofs+as+programs">proofs as programs</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/theorem">theorem</a></p> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/computational+trinitarianism">computational trinitarianism</a></strong> = <br /> <strong><a class="existingWikiWord" href="/nlab/show/propositions+as+types">propositions as types</a></strong> +<strong><a class="existingWikiWord" href="/nlab/show/programs+as+proofs">programs as proofs</a></strong> +<strong><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation type theory/category theory</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/logic">logic</a></th><th><a class="existingWikiWord" href="/nlab/show/set+theory">set theory</a> (<a class="existingWikiWord" href="/nlab/show/internal+logic+of+set+theory">internal logic</a> of)</th><th><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></th><th><a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/proposition">proposition</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/set">set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/object">object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type">type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/predicate">predicate</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/family+of+sets">family of sets</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/display+morphism">display morphism</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+type">dependent type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/proof">proof</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/element">element</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/generalized+element">generalized element</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/term">term</a>/<a class="existingWikiWord" href="/nlab/show/program">program</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cut+rule">cut rule</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/composition">composition</a> of <a class="existingWikiWord" href="/nlab/show/classifying+morphisms">classifying morphisms</a> / <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of <a class="existingWikiWord" href="/nlab/show/display+maps">display maps</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/substitution">substitution</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/introduction+rule">introduction rule</a> for <a class="existingWikiWord" href="/nlab/show/implication">implication</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/counit">counit</a> for hom-tensor adjunction</td><td style="text-align: left;">lambda</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/elimination+rule">elimination rule</a> for <a class="existingWikiWord" href="/nlab/show/implication">implication</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/unit">unit</a> for hom-tensor adjunction</td><td style="text-align: left;">application</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cut+elimination">cut elimination</a> for <a class="existingWikiWord" href="/nlab/show/implication">implication</a></td><td style="text-align: left;"></td><td style="text-align: left;">one of the <a class="existingWikiWord" href="/nlab/show/zigzag+identities">zigzag identities</a> for hom-tensor adjunction</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/beta+reduction">beta reduction</a></td></tr> <tr><td style="text-align: left;">identity elimination for <a class="existingWikiWord" href="/nlab/show/implication">implication</a></td><td style="text-align: left;"></td><td style="text-align: left;">the other <a class="existingWikiWord" href="/nlab/show/zigzag+identity">zigzag identity</a> for hom-tensor adjunction</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/eta+conversion">eta conversion</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/true">true</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/singleton">singleton</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>/<a class="existingWikiWord" href="/nlab/show/%28-2%29-truncated+object">(-2)-truncated object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/h-level+0">h-level 0</a>-<a class="existingWikiWord" href="/nlab/show/type">type</a>/<a class="existingWikiWord" href="/nlab/show/unit+type">unit type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/false">false</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/empty+set">empty set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/empty+type">empty type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/proposition">proposition</a>, <a class="existingWikiWord" href="/nlab/show/truth+value">truth value</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/subsingleton">subsingleton</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/subterminal+object">subterminal object</a>/<a class="existingWikiWord" href="/nlab/show/%28-1%29-truncated+object">(-1)-truncated object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/h-proposition">h-proposition</a>, <a class="existingWikiWord" href="/nlab/show/mere+proposition">mere proposition</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/logical+conjunction">logical conjunction</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cartesian+product">cartesian product</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/product">product</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/product+type">product type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/disjunction">disjunction</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a> (<a class="existingWikiWord" href="/nlab/show/support">support</a> of)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> (<a class="existingWikiWord" href="/nlab/show/%28-1%29-truncation">(-1)-truncation</a> of)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/sum+type">sum type</a> (<a class="existingWikiWord" href="/nlab/show/bracket+type">bracket type</a> of)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/implication">implication</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/function+set">function set</a> (into <a class="existingWikiWord" href="/nlab/show/subsingleton">subsingleton</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> (into <a class="existingWikiWord" href="/nlab/show/subterminal+object">subterminal object</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/function+type">function type</a> (into <a class="existingWikiWord" href="/nlab/show/h-proposition">h-proposition</a>)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/negation">negation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/function+set">function set</a> into <a class="existingWikiWord" href="/nlab/show/empty+set">empty set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> into <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/function+type">function type</a> into <a class="existingWikiWord" href="/nlab/show/empty+type">empty type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/universal+quantification">universal quantification</a></td><td style="text-align: left;">indexed <a class="existingWikiWord" href="/nlab/show/cartesian+product">cartesian product</a> (of family of <a class="existingWikiWord" href="/nlab/show/subsingletons">subsingletons</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a> (of family of <a class="existingWikiWord" href="/nlab/show/subterminal+objects">subterminal objects</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+product+type">dependent product type</a> (of family of <a class="existingWikiWord" href="/nlab/show/h-propositions">h-propositions</a>)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/existential+quantification">existential quantification</a></td><td style="text-align: left;">indexed <a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a> (<a class="existingWikiWord" href="/nlab/show/support">support</a> of)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a> (<a class="existingWikiWord" href="/nlab/show/%28-1%29-truncation">(-1)-truncation</a> of)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+sum+type">dependent sum type</a> (<a class="existingWikiWord" href="/nlab/show/bracket+type">bracket type</a> of)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/logical+equivalence">logical equivalence</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/bijection+set">bijection set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/object+of+isomorphisms">object of isomorphisms</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/equivalence+type">equivalence type</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/support+set">support set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/support+object">support object</a>/<a class="existingWikiWord" href="/nlab/show/%28-1%29-truncation">(-1)-truncation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/propositional+truncation">propositional truncation</a>/<a class="existingWikiWord" href="/nlab/show/bracket+type">bracket type</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/n-image">n-image</a> of <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> into <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>/<a class="existingWikiWord" href="/nlab/show/n-truncation">n-truncation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/n-truncation+modality">n-truncation modality</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/equality">equality</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/diagonal+function">diagonal function</a>/<a class="existingWikiWord" href="/nlab/show/diagonal+subset">diagonal subset</a>/<a class="existingWikiWord" href="/nlab/show/diagonal+relation">diagonal relation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/identity+type">identity type</a>/<a class="existingWikiWord" href="/nlab/show/path+type">path type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/completely+presented+set">completely presented set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/set">set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/discrete+object">discrete object</a>/<a class="existingWikiWord" href="/nlab/show/0-truncated+object">0-truncated object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/h-level+2">h-level 2</a>-<a class="existingWikiWord" href="/nlab/show/type">type</a>/<a class="existingWikiWord" href="/nlab/show/set">set</a>/<a class="existingWikiWord" href="/nlab/show/h-set">h-set</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/set">set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/set">set</a> with <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28infinity%2C1%29-category">internal 0-groupoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Bishop+set">Bishop set</a>/<a class="existingWikiWord" href="/nlab/show/setoid">setoid</a> with its <a class="existingWikiWord" href="/nlab/show/pseudo-equivalence+relation">pseudo-equivalence relation</a> an actual <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/equivalence+class">equivalence class</a>/<a class="existingWikiWord" href="/nlab/show/quotient+set">quotient set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quotient">quotient</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quotient+type">quotient type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/induction">induction</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/inductive+type">inductive type</a>, <a class="existingWikiWord" href="/nlab/show/W-type">W-type</a>, <a class="existingWikiWord" href="/nlab/show/M-type">M-type</a></td></tr> <tr><td style="text-align: left;">higher <a class="existingWikiWord" href="/nlab/show/induction">induction</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-colimit">higher colimit</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/higher+inductive+type">higher inductive type</a></td></tr> <tr><td style="text-align: left;">-</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/0-truncated">0-truncated</a> <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-colimit">higher colimit</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quotient+inductive+type">quotient inductive type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coinduction">coinduction</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/limit">limit</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coinductive+type">coinductive type</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/preset">preset</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type">type</a> without <a class="existingWikiWord" href="/nlab/show/identity+types">identity types</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/set">set</a> of <a class="existingWikiWord" href="/nlab/show/truth+values">truth values</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+of+propositions">type of propositions</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/domain+of+discourse">domain of discourse</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/universe">universe</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/object+classifier">object classifier</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+universe">type universe</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/modality">modality</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/closure+operator">closure operator</a>, (<a class="existingWikiWord" href="/nlab/show/idempotent+monad">idempotent</a>) <a class="existingWikiWord" href="/nlab/show/monad">monad</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/modal+type+theory">modal type theory</a>, <a class="existingWikiWord" href="/nlab/show/monad+%28in+computer+science%29">monad (in computer science)</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a></td><td style="text-align: left;"></td><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric</a>, <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed</a>) <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+type+theory">linear type theory</a>/<a class="existingWikiWord" href="/nlab/show/quantum+computation">quantum computation</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/proof+net">proof net</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/string+diagram">string diagram</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quantum+circuit">quantum circuit</a></td></tr> <tr><td style="text-align: left;">(absence of) <a class="existingWikiWord" href="/nlab/show/contraction+rule">contraction rule</a></td><td style="text-align: left;"></td><td style="text-align: left;">(absence of) <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/no-cloning+theorem">no-cloning theorem</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/synthetic+mathematics">synthetic mathematics</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/domain+specific+embedded+programming+language">domain specific embedded programming language</a></td></tr> </tbody></table> </div> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+levels">homotopy levels</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-type+theory">2-type theory</a>, <a class="existingWikiWord" href="/michaelshulman/show/2-categorical+logic">2-categorical logic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory+-+contents">homotopy type theory - contents</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/univalence">univalence</a>, <a class="existingWikiWord" href="/nlab/show/function+extensionality">function extensionality</a>, <a class="existingWikiWord" href="/nlab/show/internal+logic+of+an+%28%E2%88%9E%2C1%29-topos">internal logic of an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+type+theory">cohesive homotopy type theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/directed+homotopy+type+theory">directed homotopy type theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/HoTT+methods+for+homotopy+theorists">HoTT methods for homotopy theorists</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/semantics">semantics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a>, <a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/display+map">display map</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic+of+a+topos">internal logic of a topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Mitchell-Benabou+language">Mitchell-Benabou language</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kripke-Joyal+semantics">Kripke-Joyal semantics</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic+of+an+%28%E2%88%9E%2C1%29-topos">internal logic of an (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/type-theoretic+model+category">type-theoretic model category</a></li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/type+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#in_the_foundations_of_mathematics'>In the foundations of mathematics</a></li> <li><a href='#description'>Description</a></li> <ul> <li><a href='#judgments_for_types_and_terms'>Judgments for types and terms</a></li> </ul> <li><a href='#properties'>Properties</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p><em>Dependent type theory</em> is the flavor of <a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a> that admits <em><a class="existingWikiWord" href="/nlab/show/dependent+types">dependent types</a></em>.</p> <p>Its <a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a> is in <a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+categories">locally cartesian closed categories</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, where a <a class="existingWikiWord" href="/nlab/show/dependent+type">dependent type</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>X</mi><mo>⊢</mo><mi>E</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><annotation encoding="application/x-tex"> x : X \vdash E(x) \; \mathrm{type} </annotation></semantics></math></div> <p>is interpreted as a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">E \to X</annotation></semantics></math>, hence an <a class="existingWikiWord" href="/nlab/show/object">object</a> in the <a class="existingWikiWord" href="/nlab/show/slice+category">slice category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">C_{/X}</annotation></semantics></math>.</p> <p>Then change of <a class="existingWikiWord" href="/nlab/show/context">context</a> corresponds to <a class="existingWikiWord" href="/nlab/show/base+change">base change</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>. See also <em><a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a></em> and <em><a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a></em>.</p> <p>Dependent type systems are heavily used for <em><a class="existingWikiWord" href="/nlab/show/certified+programming">software certification</a></em>.</p> <h2 id="in_the_foundations_of_mathematics">In the foundations of mathematics</h2> <p>Dependent type theory itself support various <a class="existingWikiWord" href="/nlab/show/foundations+of+mathematics">foundations of mathematics</a> via the <a class="existingWikiWord" href="/nlab/show/propositions+as+some+types">propositions as some types</a> interpretation of <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a>, where propositions are the types where every two elements are equal</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">isProp</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≔</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>y</mi><mo>:</mo><mi>A</mi></mrow></munder><mi>x</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>y</mi></mrow><annotation encoding="application/x-tex">\mathrm{isProp}(A) \coloneqq \prod_{x:A} \prod_{y:A} x =_A y</annotation></semantics></math></div> <p>Suppose that a dependent type theory has a separate <a class="existingWikiWord" href="/nlab/show/type">type</a> <a class="existingWikiWord" href="/nlab/show/judgment">judgment</a> as well as <a class="existingWikiWord" href="/nlab/show/dependent+product+types">dependent product types</a>, <a class="existingWikiWord" href="/nlab/show/dependent+sum+types">dependent sum types</a>, <a class="existingWikiWord" href="/nlab/show/identity+types">identity types</a>, <a class="existingWikiWord" href="/nlab/show/weak+function+extensionality">weak function extensionality</a>, <a class="existingWikiWord" href="/nlab/show/propositional+truncations">propositional truncations</a>, <a class="existingWikiWord" href="/nlab/show/empty+type">empty type</a>, <a class="existingWikiWord" href="/nlab/show/unit+type">unit type</a>, <a class="existingWikiWord" href="/nlab/show/sum+types">sum types</a>. All the operations in <a class="existingWikiWord" href="/nlab/show/predicate+logic">predicate logic</a> are derivable from said type formers:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+quantification">universal quantification</a> is the dependent product type of families of propositions due to weak function extensionality</p> </li> <li> <p>similarly, <a class="existingWikiWord" href="/nlab/show/implication">implication</a> is the function type of two propositions due to weak function extensionality, and function types are just dependent product types of a constant family of types</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/logical+conjunction">logical conjunction</a> is the product type of two propositions, and product types are just dependent sum types of a constant family of types</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/existential+quantification">existential quantification</a> is the propositional truncation of dependent sum types</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/logical+disjunction">logical disjunction</a> is the propositional truncation of sum types</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/falsehood">falsehood</a> is the empty type</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/truth">truth</a> is the unit type</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/excluded+middle">excluded middle</a> and the <a class="existingWikiWord" href="/nlab/show/law+of+double+negation">law of double negation</a> is stated as an <a class="existingWikiWord" href="/nlab/show/inference+rule">inference rule</a> about propositions</p> </li> </ul> <p>Then</p> <ul> <li> <p>One can add a <a class="existingWikiWord" href="/nlab/show/cumulative+hierarchy">cumulative hierarchy</a> to the dependent type theory and work entirely in the cumulative hierarchy for <a class="existingWikiWord" href="/nlab/show/material+set+theory">material set theory</a></p> </li> <li> <p>One can add a <a class="existingWikiWord" href="/nlab/show/category+of+sets">category of sets</a> to the dependent type theory and work entirely in the category of sets for <a class="existingWikiWord" href="/nlab/show/structural+set+theory">structural set theory</a></p> </li> <li> <p>One can add a <a class="existingWikiWord" href="/nlab/show/type+universe">type universe</a> satisfying certain <a class="existingWikiWord" href="/nlab/show/axioms">axioms</a> and <a class="existingWikiWord" href="/nlab/show/axiom+schemata">axiom schemata</a>, such as <a class="existingWikiWord" href="/nlab/show/universe+extensionality">universe extensionality</a>, closure under identity types, closure under dependent sum types, closure under dependent product types, <a class="existingWikiWord" href="/nlab/show/propositional+resizing">propositional resizing</a>, <a class="existingWikiWord" href="/nlab/show/axiom+of+replacement">replacement</a>, <a class="existingWikiWord" href="/nlab/show/axiom+of+infinity">infinity</a>, and <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">choice</a>, to the dependent type theory and work entirely in the universe for <a class="existingWikiWord" href="/nlab/show/univalent+type+theory">univalent type theory</a> or <a class="existingWikiWord" href="/nlab/show/univalent+foundations">univalent foundations</a>. Adding internal universe types as <a class="existingWikiWord" href="/nlab/show/small">small</a> <a class="existingWikiWord" href="/nlab/show/object+classifiers">object classifiers</a> as well as all <a class="existingWikiWord" href="/nlab/show/higher+inductive+types">higher inductive</a> and <a class="existingWikiWord" href="/nlab/show/coinductive+types">coinductive types</a> to the universe results in <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>.</p> </li> <li> <p>One can add a <a class="existingWikiWord" href="/nlab/show/Russell+type+of+all+propositions">Russell type of all propositions</a> and a <a class="existingWikiWord" href="/nlab/show/natural+numbers+type">natural numbers type</a> and work in the dependent type theory itself for <a class="existingWikiWord" href="/nlab/show/higher-order+logic">higher-order logic</a>.</p> </li> </ul> <h2 id="description">Description</h2> <h3 id="judgments_for_types_and_terms">Judgments for types and terms</h3> <div> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a></th><th><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/syntax">syntax</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/semantics">semantics</a></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/judgment">judgment</a></strong></td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/diagram">diagram</a></strong></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type">type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/object">object</a> in <a class="existingWikiWord" href="/nlab/show/category">category</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊢</mo><mspace width="thickmathspace"></mspace><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><annotation encoding="application/x-tex">\vdash\; A \; \mathrm{type}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">A \in \mathcal{C}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/term">term</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/element">element</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊢</mo><mspace width="thickmathspace"></mspace><mi>a</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\vdash\; a \colon A</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mover><mo>→</mo><mi>a</mi></mover><mi>A</mi></mrow><annotation encoding="application/x-tex">* \stackrel{a}{\to} A</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+type">dependent type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/object">object</a> in <a class="existingWikiWord" href="/nlab/show/slice+category">slice category</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mspace width="thickmathspace"></mspace><mo>⊢</mo><mspace width="thickmathspace"></mspace><mi>A</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><annotation encoding="application/x-tex">x \colon X \;\vdash\; A(x) \; \mathrm{type}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd></mtr></mtable></mrow><mo>∈</mo><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\array{A \\ \downarrow \\ X} \in \mathcal{C}_{/X}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/term+in+context">term in context</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/generalized+elements">generalized elements</a>/<a class="existingWikiWord" href="/nlab/show/element">element</a> in <a class="existingWikiWord" href="/nlab/show/slice+category">slice category</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mspace width="thickmathspace"></mspace><mo>⊢</mo><mspace width="thickmathspace"></mspace><mi>a</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x \colon X \;\vdash \; a(x)\colon A(x)</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>a</mi></mover></mtd> <mtd></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>id</mi> <mi>X</mi></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{X &&\stackrel{a}{\to}&& A \\ & {}_{\mathllap{id_X}}\searrow && \swarrow_{\mathrlap{}} \\ && X}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mspace width="thickmathspace"></mspace><mo>⊢</mo><mspace width="thickmathspace"></mspace><mi>a</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">x \colon X \;\vdash \; a(x)\colon A</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>X</mi></msub><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd></mtd> <mtd><mi>X</mi><mo>×</mo><mi>A</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>id</mi> <mi>X</mi></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{X &&\stackrel{(id_X,a)}{\to}&& X \times A \\ & {}_{\mathllap{id_X}}\searrow && \swarrow_{\mathrlap{p_1}} \\ && X}</annotation></semantics></math></td></tr> </tbody></table> </div> <h2 id="properties">Properties</h2> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p>The <a class="existingWikiWord" href="/nlab/show/functors">functors</a></p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cont</mi></mrow><annotation encoding="application/x-tex">Cont</annotation></semantics></math>, that form a <a class="existingWikiWord" href="/nlab/show/category+of+contexts">category of contexts</a> of a <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Lang</mi></mrow><annotation encoding="application/x-tex">Lang</annotation></semantics></math> that forms the <a class="existingWikiWord" href="/nlab/show/internal+language">internal language</a> of a <a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+category">locally cartesian closed category</a></p> </li> </ul> <p>constitute an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>DependentTypeTheories</mi><mover><munder><mo>→</mo><mi>Cont</mi></munder><mover><mo>←</mo><mi>Lang</mi></mover></mover><mi>LocallyCartesianClosedCategories</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> DependentTypeTheories \stackrel{\overset{Lang}{\leftarrow}}{\underset{Cont}{\to}} LocallyCartesianClosedCategories \,. </annotation></semantics></math></div></div> <p>This (<a href="#Seely">Seely, theorem 6.3</a>). It is somewhat more complicated than this, because we need to strictify the category theory to match the category theory; see <a class="existingWikiWord" href="/nlab/show/categorical+model+of+dependent+types">categorical model of dependent types</a>. For a more detailed discussion see at <em><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></em>.</p> <h2 id="examples">Examples</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/objective+type+theory">objective type theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/weak+type+theory">weak type theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/book+HoTT">book HoTT</a></li> <li><a class="existingWikiWord" href="/nlab/show/Martin-L%C3%B6f+type+theory">Martin-Löf type theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/cubical+type+theory">cubical type theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/higher+observational+type+theory">higher observational type theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/univalent+type+theory">univalent type theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/propositional+logic+as+a+dependent+type+theory">propositional logic as a dependent type theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/impredicative+dependent+type+theory">impredicative dependent type theory</a></li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+semantics+of+dependent+types">categorical semantics of dependent types</a></p> <p><a class="existingWikiWord" href="/nlab/show/relation+between+category+theory+and+type+theory">relation between category theory and type theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+dependent+type+theory">linear dependent type theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/set+theory+and+dependent+type+theory">set theory and dependent type theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dependent+type+theoretic+methods+in+natural+language+semantics">dependent type theoretic methods in natural language semantics</a></p> </li> <li> <p><span class="newWikiWord">spartan type theory<a href="/nlab/new/spartan+type+theory">?</a></span></p> </li> </ul> <h2 id="references">References</h2> <p>For original references see at <em><a class="existingWikiWord" href="/nlab/show/Martin-L%C3%B6f+dependent+type+theory">Martin-Löf dependent type theory</a></em>, such as:</p> <ul> <li id="Hofmann95"><a class="existingWikiWord" href="/nlab/show/Martin+Hofmann">Martin Hofmann</a>, <em>Syntax and semantics of dependent types</em>, Chapter 2 in: <em>Extensional concepts in intensional type theory</em>, Ph.D. thesis, University of Edinburgh (1995), Distinguished Dissertations, Springer (1997) [<a href="http://www.lfcs.inf.ed.ac.uk/reports/95/ECS-LFCS-95-327/">ECS-LFCS-95-327</a>, <a class="existingWikiWord" href="/nlab/files/HofmannExtensionalIntensionalTypeTheory.pdf" title="pdf">pdf</a>, <a href="https://doi.org/10.1007/978-1-4471-0963-1">doi:10.1007/978-1-4471-0963-1</a>]</li> </ul> <p>also published as:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Martin+Hofmann">Martin Hofmann</a>, <em>Syntax and semantics of dependent types</em>, in <em>Semantics and logics of computation</em>, Publ. Newton Inst. <strong>14</strong>, Cambridge Univ. Press (1997) 79-130 [<a href="https://doi.org/10.1017/CBO9780511526619.004">doi:10.1017/CBO9780511526619.004</a>]</li> </ul> <p>Gentle exposition of the basic principles:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Thomas+Kehrenberg">Thomas Kehrenberg</a>, <em>Introduction to dependent type theory</em> (2022-23) [<a href="https://tm.kehrenberg.net/series/introduction-to-dependent-type-theory/">blog series</a>]</li> </ul> <p>Introductory accounts:</p> <ul> <li id="Thompson91"> <p><a class="existingWikiWord" href="/nlab/show/Simon+Thompson">Simon Thompson</a>, §6.3 in: <em><a class="existingWikiWord" href="/nlab/show/Type+Theory+and+Functional+Programming">Type Theory and Functional Programming</a></em>, Addison-Wesley (1991) [ISBN:0-201-41667-0, <a href="http://www.cs.kent.ac.uk/people/staff/sjt/TTFP">webpage</a>, <a href="http://www.cs.kent.ac.uk/people/staff/sjt/TTFP/ttfp.pdf">pdf</a>]</p> </li> <li id="Jacobs98"> <p><a class="existingWikiWord" href="/nlab/show/Bart+Jacobs">Bart Jacobs</a>, Chapter 10 in: <em>Categorical Logic and Type Theory</em>, Studies in Logic and the Foundations of Mathematics <strong>141</strong>, Elsevier (1998) [<a href="https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/141">ISBN:978-0-444-50170-7</a>, <a href="https://people.mpi-sws.org/~dreyer/courses/catlogic/jacobs.pdf">pdf</a>, <a href="http://www.cs.ru.nl/B.Jacobs/CLT/bookinfo.html">webpage</a>]</p> <blockquote> <p>(emphasis on the <a class="existingWikiWord" href="/nlab/show/categorical+model+of+dependent+types">categorical model of dependent types</a>)</p> </blockquote> </li> </ul> <p>Introduction with parallel details on using <a class="existingWikiWord" href="/nlab/show/proof+assistants">proof assistants</a>:</p> <p>for <a class="existingWikiWord" href="/nlab/show/Coq">Coq</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Adam+Chlipala">Adam Chlipala</a>, <em>Certified programming with dependent types</em>, MIT Press 2013 [<a href="https://mitpress.mit.edu/books/certified-programming-dependent-types">ISBN:9780262026659</a>, <a href="http://adam.chlipala.net/cpdt/cpdt.pdf">pdf</a>, <a href="http://adam.chlipala.net/cpdt/">book webpage</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Th%C3%A9o+Winterhalter">Théo Winterhalter</a>, <em>Formalisation and Meta-Theory of Type Theory</em>, Nantes (2020) [<a href="https://github.com/TheoWinterhalter/phd-thesis/releases/download/v1.2.1/TheoWinterhalter-PhD-v1.2.1.pdf">pdf</a>, <a href="https://github.com/TheoWinterhalter/phd-thesis">github</a>]</p> </li> </ul> <p>for <a class="existingWikiWord" href="/nlab/show/Agda">Agda</a>:</p> <ul> <li id="Norell08"> <p><a class="existingWikiWord" href="/nlab/show/Ulf+Norell">Ulf Norell</a>, <em>Dependently Typed Programming in Agda</em>, p. 230-266 in: <em>Advanced Functional Programming</em> AFP 2008. Lecture Notes in Computer Science <strong>5832</strong> (2009) [<a href="https://doi.org/10.1007/978-3-642-04652-0_5">doi:10.1007/978-3-642-04652-0_5</a>, <a href="https://www.cse.chalmers.se/~ulfn/papers/afp08/tutorial.pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Agda">Agda</a> Tutorial: <em>Introduction to dependent type theory</em> (<a href="http://ocvs.cfv.jp/Agda/tutorial/node128.html">webpage</a>)</p> </li> </ul> <p>Original discussion of dependent type theory as the <a class="existingWikiWord" href="/nlab/show/internal+language">internal language</a> of <a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+categories">locally cartesian closed categories</a> is in</p> <ul> <li id="Seely"><a class="existingWikiWord" href="/nlab/show/R.+A.+G.+Seely">R. A. G. Seely</a>, <em>Locally cartesian closed categories and type theory</em>, Math. Proc. Camb. Phil. Soc. (1984) 95 (<a href="http://www.math.mcgill.ca/rags/LCCC/LCCC.pdf">pdf</a>)</li> </ul> <p>A formal definition of dependent type theories beyond <a class="existingWikiWord" href="/nlab/show/Martin-L%C3%B6f+dependent+type+theory">Martin-Löf dependent type theory</a>:</p> <ul> <li id="BauerHaselwarterLumsdaine20"><a class="existingWikiWord" href="/nlab/show/Andrej+Bauer">Andrej Bauer</a>, <a class="existingWikiWord" href="/nlab/show/Philipp+G.+Haselwarter">Philipp G. Haselwarter</a>, <a class="existingWikiWord" href="/nlab/show/Peter+LeFanu+Lumsdaine">Peter LeFanu Lumsdaine</a>, <em>A general definition of dependent type theories</em> [<a href="https://arxiv.org/abs/2009.05539">arXiv:2009.05539</a>]</li> </ul> <p id="CSystemsReferences"> On (<a class="existingWikiWord" href="/nlab/show/essentially+algebraic+theory">essentially algebraic</a>) formulations of dependent type theory (see <a href="categorical+model+of+dependent+types#ContextualCategoriesOrCSystems">here</a> at <em><a class="existingWikiWord" href="/nlab/show/categorical+models+of+dependent+type+theory">categorical models of dependent type theory</a></em>):</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Egbert+Rijke">Egbert Rijke</a>, <em>An algebraic formulation of dependent type theory</em>, (<a href="https://groups.google.com/forum/#!topic/homotopytypetheory/OraMqbnCYy8/discussion">mailing list discussion</a>)</li> <li>Vladimir Voevodsky, <em>B-systems</em>, (<a href="http://arxiv.org/abs/1410.5389">arXiv:1410.5389</a>)</li> <li>Vladimir Voevodsky, <em>A C-system defined by a universe in a category</em>, (<a href="http://arxiv.org/abs/1409.7925">arXiv:1406.7413</a>)</li> <li>Vladimir Voevodsky, <em>C-system of a module over a monad on sets</em>, (<a href="http://arxiv.org/abs/1407.3394">arXiv:1406.7413</a>)</li> <li>Vladimir Voevodsky, <em>Subsystems and regular quotients of C-systems</em>, (<a href="http://arxiv.org/abs/1406.7413">arXiv:1406.7413</a>)</li> </ul> <p>For more see the references at <em><a class="existingWikiWord" href="/nlab/show/Martin-L%C3%B6f+dependent+type+theory">Martin-Löf dependent type theory</a></em>.</p> </body></html> </div> <div class="revisedby"> <p> Last revised on August 20, 2024 at 17:09:45. 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