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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> judgmental equality </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/11658/#Item_4" 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" 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<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> <h4 id="equality_and_equivalence">Equality and Equivalence</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/equivalence">equivalence</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equality">equality</a> (<a class="existingWikiWord" href="/nlab/show/definitional+equality">definitional</a>, <a class="existingWikiWord" href="/nlab/show/propositional+equality">propositional</a>, <a class="existingWikiWord" href="/nlab/show/computational+equality">computational</a>, <a class="existingWikiWord" href="/nlab/show/judgemental+equality">judgemental</a>, <a class="existingWikiWord" href="/nlab/show/extensional+equality">extensional</a>, <a class="existingWikiWord" href="/nlab/show/intensional+equality">intensional</a>, <a class="existingWikiWord" href="/nlab/show/decidable+equality">decidable</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/identity+type">identity type</a>, <a class="existingWikiWord" href="/nlab/show/equivalence+of+types">equivalence of types</a>, <a class="existingWikiWord" href="/nlab/show/definitional+isomorphism">definitional isomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>, <a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a>, <a class="existingWikiWord" href="/nlab/show/equivalence+in+an+%28%E2%88%9E%2C1%29-category">equivalence in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+equivalence">natural equivalence</a>, <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+equivalence">gauge equivalence</a></p> </li> <li> <p><strong>Examples.</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a>, <a class="existingWikiWord" href="/nlab/show/adjoint+equivalence">adjoint equivalence</a>, <a class="existingWikiWord" href="/nlab/show/weak+equivalence+of+internal+categories">weak equivalence of internal categories</a>, <a class="existingWikiWord" href="/nlab/show/Morita+equivalence">Morita equivalence</a>, <a class="existingWikiWord" href="/nlab/show/equivalence+of+2-categories">equivalence of 2-categories</a>, <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a></li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/principle+of+equivalence">principle of equivalence</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/univalence">univalence</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/equation">equation</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a>, <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a></p> </li> <li> <p><strong>Examples.</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+equation">linear equation</a>, <a class="existingWikiWord" href="/nlab/show/differential+equation">differential equation</a>, <a class="existingWikiWord" href="/nlab/show/ordinary+differential+equation">ordinary differential equation</a>, <a class="existingWikiWord" href="/nlab/show/critical+locus">critical locus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equation">Euler-Lagrange equation</a>, <a class="existingWikiWord" href="/nlab/show/Einstein+equation">Einstein equation</a>, <a class="existingWikiWord" href="/nlab/show/wave+equation">wave equation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Schr%C3%B6dinger+equation">Schrödinger equation</a>, <a class="existingWikiWord" href="/nlab/show/Knizhnik-Zamolodchikov+equation">Knizhnik-Zamolodchikov equation</a>, <a class="existingWikiWord" href="/nlab/show/Maurer-Cartan+equation">Maurer-Cartan equation</a>, <a class="existingWikiWord" href="/nlab/show/quantum+master+equation">quantum master equation</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Arnold+equation">Euler-Arnold equation</a>, <a class="existingWikiWord" href="/nlab/show/Fuchsian+equation">Fuchsian equation</a>, <a class="existingWikiWord" href="/nlab/show/Fokker-Planck+equation">Fokker-Planck equation</a>, <a class="existingWikiWord" href="/nlab/show/Lax+equation">Lax equation</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/equality+and+equivalence+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id='section_table_of_contents'>Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#judgmental_equality_of_terms'>Judgmental equality of terms</a></li> <ul> <li><a href='#weak_judgmental_equality'>Weak judgmental equality</a></li> <li><a href='#strict_judgmental_equality'>Strict judgmental equality</a></li> <li><a href='#in_computation_and_uniqueness_rules'>In computation and uniqueness rules</a></li> </ul> <li><a href='#judgmental_equality_of_types'>Judgmental equality of types</a></li> <ul> <li><a href='#weak_judgmental_equality_2'>Weak judgmental equality</a></li> <li><a href='#strict_judgmental_equality_2'>Strict judgmental equality</a></li> <li><a href='#congruence_rules_for_judgmental_equality_of_types'>Congruence rules for judgmental equality of types</a></li> </ul> <li><a href='#judgmental_equality_of_contexts'>Judgmental equality of contexts</a></li> <li><a href='#see_also'>See also</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>In any <a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a>, judgmental equality is the notion of <a class="existingWikiWord" href="/nlab/show/equality">equality</a> which is defined to be a <a class="existingWikiWord" href="/nlab/show/judgment">judgment</a>. Judgmental equality is most commonly used in single-level type theories like <a class="existingWikiWord" href="/nlab/show/Martin-L%C3%B6f+type+theory">Martin-Löf type theory</a> or <a class="existingWikiWord" href="/nlab/show/higher+observational+type+theory">higher observational type theory</a> for making <a class="existingWikiWord" href="/nlab/show/inductive+definitions">inductive definitions</a>, but it is also used in <a class="existingWikiWord" href="/nlab/show/cubical+type+theory">cubical type theory</a> and <a class="existingWikiWord" href="/nlab/show/simplicial+type+theory">simplicial type theory</a> to define probe shapes for <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category+theory">(infinity,1)-categorical</a> types which could not be coherently defined in vanilla dependent type theory.</p> <p>There are two different kinds of judgmental equalities</p> <ul> <li> <p>Judgmental equality of terms</p> </li> <li> <p>Judgmental equality of types, in <a class="existingWikiWord" href="/nlab/show/dependent+type+theories">dependent type theories</a> with a separate <a class="existingWikiWord" href="/nlab/show/type">type</a> <a class="existingWikiWord" href="/nlab/show/judgment">judgment</a>.</p> </li> </ul> <p>Judgmental equality of types is not necessary for <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a> with a separate type judgment. It behaves similarly to the <a class="existingWikiWord" href="/nlab/show/equality">equality</a> between <a class="existingWikiWord" href="/nlab/show/sets">sets</a> in <a class="existingWikiWord" href="/nlab/show/structural+set+theory">structural set theory</a>, and the equality between sets is not necessary for structural set theory since one could simply work with <a class="existingWikiWord" href="/nlab/show/bijections">bijections</a> or <a class="existingWikiWord" href="/nlab/show/one-to-one+correspondences">one-to-one correspondences</a> between sets. Similarly, in <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a>, one could just work with <a class="existingWikiWord" href="/nlab/show/definitional+isomorphism">definitional isomorphism</a> or some notion of <a class="existingWikiWord" href="/nlab/show/equivalence+of+types">equivalence of types</a> instead of judgmental equality of types.</p> <h2 id="judgmental_equality_of_terms">Judgmental equality of terms</h2> <p>Judgmental equality of terms is given by the following judgment:</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo>⊢</mo><mi>a</mi><mo>≡</mo><mi>a</mi><mo>′</mo><mo>:</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\Gamma \vdash a \equiv a' : A</annotation></semantics></math> - <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">a'</annotation></semantics></math> are judgmentally equal well-typed terms of type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> in context <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math>.</li> </ul> <p>There are two different notions of judgmental equality of terms which could be distinguished:</p> <ul> <li> <p>Weak judgmental equality of terms is just a shorthand for the <a class="existingWikiWord" href="/nlab/show/identity+type">identity type</a> as <a class="existingWikiWord" href="/nlab/show/typal+equality">typal equality</a></p> </li> <li> <p>Strict judgmental equality is additional structure on types which gives every type the structure of a <a class="existingWikiWord" href="/nlab/show/set">set</a> in addition to the <a class="existingWikiWord" href="/nlab/show/infinity-groupoid"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-groupoidal</a> structure on a type from the <a class="existingWikiWord" href="/nlab/show/identity+type">identity type</a>.</p> </li> </ul> <p>Strict judgmental equalities of terms are used in most <a class="existingWikiWord" href="/nlab/show/dependent+type+theories">dependent type theories</a>. Weak judgmental equalities of terms can be used in <a class="existingWikiWord" href="/nlab/show/weak+type+theories">weak type theories</a>, where a direct translation of the <a class="existingWikiWord" href="/nlab/show/inference+rules">inference rules</a> of the types in <a class="existingWikiWord" href="/nlab/show/Martin-L%C3%B6f+type+theory">Martin-Löf type theory</a> results in a weak version of Martin-Löf type theory.</p> <p>Judgmental equality of terms can be contrasted with <a class="existingWikiWord" href="/nlab/show/propositional+equality">propositional equality</a> of terms, where equality is a <a class="existingWikiWord" href="/nlab/show/proposition">proposition</a> in the sense of <a class="existingWikiWord" href="/nlab/show/first-order+logic">first-order logic</a>, and <a class="existingWikiWord" href="/nlab/show/typal+equality">typal equality</a> of terms, where equality is a <a class="existingWikiWord" href="/nlab/show/type">type</a>.</p> <h3 id="weak_judgmental_equality">Weak judgmental equality</h3> <p>Weak judgmental equality of terms is simply given by a reflection rule into the <a class="existingWikiWord" href="/nlab/show/identity+type">identity type</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>a</mi><mo>≡</mo><mi>a</mi><mo>′</mo><mo>:</mo><mi>A</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><msub><mi>δ</mi> <mrow><mi>a</mi><mo>,</mo><mi>a</mi><mo>′</mo></mrow></msub><mo>:</mo><mi>a</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>a</mi><mo>′</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash a \equiv a':A}{\Gamma \vdash \delta_{a, a'}:a =_A a'}</annotation></semantics></math></div> <h3 id="strict_judgmental_equality">Strict judgmental equality</h3> <p>Strict judgmental equality is an equivalence relation:</p> <ul> <li>Reflexivity of judgmental equality</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>a</mi><mo>:</mo><mi>A</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><mi>a</mi><mo>≡</mo><mi>a</mi><mo>:</mo><mi>A</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a:A}{\Gamma \vdash a \equiv a:A}</annotation></semantics></math></div> <ul> <li> <p>Symmetry of judgmental equality</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>a</mi><mo>≡</mo><mi>b</mi><mo>:</mo><mi>A</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><mi>b</mi><mo>≡</mo><mi>a</mi><mo>:</mo><mi>A</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a \equiv b:A}{\Gamma \vdash b \equiv a:A}</annotation></semantics></math></div></li> <li> <p>Transitivity of judgmental equality</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>a</mi><mo>≡</mo><mi>b</mi><mo>:</mo><mi>A</mi><mspace width="1em"></mspace><mi>b</mi><mo>≡</mo><mi>c</mi><mo>:</mo><mi>A</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><mi>a</mi><mo>≡</mo><mi>c</mi><mo>:</mo><mi>A</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a \equiv b:A \quad b \equiv c:A }{\Gamma \vdash a \equiv c:A}</annotation></semantics></math></div></li> </ul> <p>In addition, strict judgmental equality of terms has congruence rules for substitution, the <a class="existingWikiWord" href="/nlab/show/principle+of+substitution">principle of substitution</a>:</p> <ul> <li>Principle of substitution for judgmentally equal terms:<div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>a</mi><mo>≡</mo><mi>b</mi><mo>:</mo><mi>A</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>,</mo><mi>Δ</mi><mo>⊢</mo><mi>c</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><mi>B</mi></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>Δ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>⊢</mo><mi>c</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>c</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>:</mo><mi>B</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash a \equiv b : A \quad \Gamma, x:A, \Delta \vdash c(x):B}{\Gamma, \Delta(a) \vdash c(a) \equiv c(b): B}</annotation></semantics></math></div></li> </ul> <p>If there is a separate <a class="existingWikiWord" href="/nlab/show/type">type</a> <a class="existingWikiWord" href="/nlab/show/judgment">judgment</a>, then there is also a separate rule for the principle of substitution into type families.</p> <p>If one has judgmental equality of types, then the principle of substitution into type families is given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>a</mi><mo>≡</mo><mi>b</mi><mo>:</mo><mi>A</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>,</mo><mi>Δ</mi><mo>⊢</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>Δ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>⊢</mo><mi>B</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>B</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a \equiv b : A \quad \Gamma, x:A, \Delta \vdash B(x) \; \mathrm{type}}{\Gamma, \Delta(a) \vdash B(a) \equiv B(b) \; \mathrm{type}}</annotation></semantics></math></div> <p>This implies the reflection rule of weak judgmental equalities because one could derive the following rule:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>a</mi><mo>≡</mo><mi>b</mi><mo>:</mo><mi>A</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><msub><mi mathvariant="normal">refl</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>:</mo><mi>a</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>b</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash a \equiv b:A}{\Gamma \vdash \mathrm{refl}_A(a):a =_A b}</annotation></semantics></math></div> <p>Otherwise, the principle of substitution into type families is given by <a class="existingWikiWord" href="/nlab/show/definitional+transport">definitional transport</a> across judgmental equality as <a class="existingWikiWord" href="/nlab/show/explicit+conversion">explicit conversion</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>a</mi><mo>≡</mo><mi>b</mi><mo>:</mo><mi>A</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>,</mo><mi>Δ</mi><mo>⊢</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>Δ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>⊢</mo><msubsup><mi mathvariant="normal">tr</mi> <mrow><mi>B</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow> <mrow><mi>a</mi><mo>≡</mo><mi>b</mi></mrow></msubsup><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>≅</mo><mi>B</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a \equiv b : A \quad \Gamma, x:A, \Delta \vdash B(x) \; \mathrm{type}}{\Gamma, \Delta(a) \vdash \mathrm{tr}_{B(-)}^{a \equiv b}:B(a) \cong B(b)}</annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>≅</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A \cong B</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/definitional+isomorphism+type">definitional isomorphism type</a> defined using <a class="existingWikiWord" href="/nlab/show/natural+deduction">natural deduction</a> <a class="existingWikiWord" href="/nlab/show/inference+rules">inference rules</a>. If one doesn’t have a type of <a class="existingWikiWord" href="/nlab/show/definitional+isomorphisms">definitional isomorphisms</a>, one could define it by components</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>a</mi><mo>≡</mo><mi>b</mi><mo>:</mo><mi>A</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>,</mo><mi>Δ</mi><mo>⊢</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>y</mi><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Δ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>⊢</mo><msubsup><mi mathvariant="normal">tr</mi> <mrow><mi>B</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow> <mrow><mi>a</mi><mo>≡</mo><mi>b</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a \equiv b : A \quad \Gamma, x:A, \Delta \vdash B(x) \; \mathrm{type}}{\Gamma, y:B(a), \Delta(a) \vdash \mathrm{tr}_{B(-)}^{a \equiv b}(y):B(b)}</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>a</mi><mo>:</mo><mi>A</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>,</mo><mi>Δ</mi><mo>⊢</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>y</mi><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Δ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>⊢</mo><msubsup><mi mathvariant="normal">tr</mi> <mrow><mi>B</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow> <mrow><mi>a</mi><mo>≡</mo><mi>a</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>y</mi><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a: A \quad \Gamma, x:A, \Delta \vdash B(x) \; \mathrm{type}}{\Gamma, y:B(a), \Delta(a) \vdash \mathrm{tr}_{B(-)}^{a \equiv a}(y) \equiv y:B(a)}</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>a</mi><mo>≡</mo><mi>b</mi><mo>:</mo><mi>A</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>,</mo><mi>Δ</mi><mo>⊢</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>y</mi><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Δ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>⊢</mo><msubsup><mi mathvariant="normal">tr</mi> <mrow><mi>B</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow> <mrow><mi>b</mi><mo>≡</mo><mi>a</mi></mrow></msubsup><mo stretchy="false">(</mo><msubsup><mi mathvariant="normal">tr</mi> <mrow><mi>B</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow> <mrow><mi>a</mi><mo>≡</mo><mi>b</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><mi>y</mi><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a \equiv b : A \quad \Gamma, x:A, \Delta \vdash B(x) \; \mathrm{type}}{\Gamma, y:B(a), \Delta(a) \vdash \mathrm{tr}_{B(-)}^{b \equiv a}(\mathrm{tr}_{B(-)}^{a \equiv b}(y)) \equiv y:B(a)}</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>a</mi><mo>≡</mo><mi>b</mi><mo>:</mo><mi>A</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>b</mi><mo>≡</mo><mi>c</mi><mo>:</mo><mi>A</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>,</mo><mi>Δ</mi><mo>⊢</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>y</mi><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Δ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>⊢</mo><msubsup><mi mathvariant="normal">tr</mi> <mrow><mi>B</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow> <mrow><mi>b</mi><mo>≡</mo><mi>c</mi></mrow></msubsup><mo stretchy="false">(</mo><msubsup><mi mathvariant="normal">tr</mi> <mrow><mi>B</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow> <mrow><mi>a</mi><mo>≡</mo><mi>b</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><msubsup><mi mathvariant="normal">tr</mi> <mrow><mi>B</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow> <mrow><mi>a</mi><mo>≡</mo><mi>c</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a \equiv b : A \quad \Gamma \vdash b \equiv c : A \quad \Gamma, x:A, \Delta \vdash B(x) \; \mathrm{type}}{\Gamma, y:B(a), \Delta(a) \vdash \mathrm{tr}_{B(-)}^{b \equiv c}(\mathrm{tr}_{B(-)}^{a \equiv b}(y)) \equiv \mathrm{tr}_{B(-)}^{a \equiv c}(y):B(c)}</annotation></semantics></math></div> <p>This shows that transport across judgmental equality forms a <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a>.</p> <p>Either way, this also implies the reflection rule of weak judgmental equalities because one could derive the following rule</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>a</mi><mo>≡</mo><mi>b</mi><mo>:</mo><mi>A</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><msubsup><mi mathvariant="normal">tr</mi> <mrow><mi>a</mi><msub><mo>=</mo> <mi>A</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow> <mrow><mi>a</mi><mo>≡</mo><mi>b</mi></mrow></msubsup><mo stretchy="false">(</mo><msub><mi mathvariant="normal">refl</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><mi>a</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>b</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash a \equiv b:A}{\Gamma \vdash \mathrm{tr}_{a =_A (-)}^{a \equiv b}(\mathrm{refl}_A(a)):a =_A b}</annotation></semantics></math></div> <p>Similarly, for a term <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c(x):B(x)</annotation></semantics></math> dependent upon <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">x:A</annotation></semantics></math>, if one has judgmental equality of types, then the principle of substitution across <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c(x)</annotation></semantics></math> is given by the rule:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>a</mi><mo>≡</mo><mi>b</mi><mo>:</mo><mi>A</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>,</mo><mi>Δ</mi><mo>⊢</mo><mi>c</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>Δ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>⊢</mo><mi>c</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>c</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a \equiv b : A \quad \Gamma, x:A, \Delta \vdash c(x):B(x)}{\Gamma, \Delta(b) \vdash c(a) \equiv c(b):B(b)}</annotation></semantics></math></div> <p>Otherwise, it is given by a judgmental version of <a class="existingWikiWord" href="/nlab/show/function+application+to+identifications">function application to identifications</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>a</mi><mo>≡</mo><mi>b</mi><mo>:</mo><mi>A</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>,</mo><mi>Δ</mi><mo>⊢</mo><mi>c</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>Δ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>⊢</mo><msubsup><mi mathvariant="normal">tr</mi> <mrow><mi>B</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow> <mrow><mi>a</mi><mo>≡</mo><mi>b</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><mi>c</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a \equiv b : A \quad \Gamma, x:A, \Delta \vdash c(x):B(x)}{\Gamma, \Delta(b) \vdash \mathrm{tr}_{B(-)}^{a \equiv b}(c(a)) \equiv c(b):B(b)}</annotation></semantics></math></div> <h3 id="in_computation_and_uniqueness_rules">In computation and uniqueness rules</h3> <p>Judgmental equality of terms can be used in the <a class="existingWikiWord" href="/nlab/show/computation+rules">computation rules</a> and <a class="existingWikiWord" href="/nlab/show/uniqueness+rules">uniqueness rules</a> of types:</p> <ul> <li>Computation rules for dependent product types:</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>b</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>a</mi><mo>:</mo><mi>A</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><mi>λ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo stretchy="false">)</mo><mo>.</mo><mi>b</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>b</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma, x:A \vdash b(x):B(x) \quad \Gamma \vdash a:A}{\Gamma \vdash \lambda(x:A).b(x)(a) \equiv b(a):B(a)}</annotation></semantics></math></div> <ul> <li>Uniqueness rules for dependent product types:</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>f</mi><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>Γ</mi><mo>⊢</mo><mi>f</mi><mo>≡</mo><mi>λ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>.</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</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><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash f:\prod_{x:A} B(x)}{\Gamma \vdash f \equiv \lambda(x).f(x):\prod_{x:A} B(x)}</annotation></semantics></math></div> <ul> <li>Computation rules for negative dependent sum types:</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>a</mi><mo>:</mo><mi>A</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>b</mi><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><mrow><mi>Γ</mi><mo>⊢</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>a</mi><mo>:</mo><mi>A</mi></mrow></mfrac><mspace width="2em"></mspace><mfrac><mrow><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>a</mi><mo>:</mo><mi>A</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>b</mi><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><mrow><mi>Γ</mi><mo>⊢</mo><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>b</mi><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash b:B(a)}{\Gamma \vdash \pi_1(a, b) \equiv a:A} \qquad \frac{\Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash b:B(a)}{\Gamma \vdash \pi_2(a, b) \equiv b:B(a)}</annotation></semantics></math></div> <p>If one does not have judgmental equality of types, then one would have to use transport across judgmental equality for the second computation rule:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>a</mi><mo>:</mo><mi>A</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>b</mi><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><mrow><mi>Γ</mi><mo>⊢</mo><msubsup><mi mathvariant="normal">tr</mi> <mrow><mi>B</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow> <mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>a</mi></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><mi>b</mi><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash b:B(a)}{\Gamma \vdash \mathrm{tr}_{B(-)}^{\pi_1(a, b) \equiv a}(\pi_2(a, b)) \equiv b:B(a)}</annotation></semantics></math></div> <ul> <li>Uniqueness rules for negative dependent sum types:</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>z</mi><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>Γ</mi><mo>⊢</mo><mi>z</mi><mo>≡</mo><mo stretchy="false">(</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash z:\sum_{x:A} B(x)}{\Gamma \vdash z \equiv (\pi_1(z), \pi_2(z)):\sum_{x:A} B(x)}</annotation></semantics></math></div> <ul> <li>Computation rules for identity types:</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>,</mo><mi>a</mi><mo>:</mo><mi>A</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>A</mi><mo>,</mo><mi>p</mi><mo>:</mo><mi>a</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>b</mi><mo>⊢</mo><mi>C</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>t</mi><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>c</mi><mo>:</mo><mi>A</mi></mrow></munder><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo>,</mo><msub><mi mathvariant="normal">refl</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>c</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>J</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi mathvariant="normal">refl</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><mi>t</mi><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo>,</mo><msub><mi mathvariant="normal">refl</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma, a:A, b:A, p:a =_A b \vdash C(a, b, p) \; \mathrm{type} \quad \Gamma \vdash t:\prod_{c:A} C(c, c, \mathrm{refl}_A(c))}{\Gamma, c:A \vdash J(t, c, c, \mathrm{refl}(c)) \equiv t:C(c, c, \mathrm{refl}_A(c))}</annotation></semantics></math></div> <h2 id="judgmental_equality_of_types">Judgmental equality of types</h2> <p>In <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a> with a separate <a class="existingWikiWord" href="/nlab/show/type">type</a> <a class="existingWikiWord" href="/nlab/show/judgment">judgment</a>, judgmental equality of types is given by the following judgment:</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mo>≡</mo><mi>A</mi><mo>′</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><annotation encoding="application/x-tex">\Gamma \vdash A \equiv A' \; \mathrm{type}</annotation></semantics></math> - <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">A'</annotation></semantics></math> are judgmentally equal well-typed types in context <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math>.</li> </ul> <p>There are two different notions of judgmental equality of types which could be distinguished:</p> <ul> <li> <p>Weak judgmental equality of types is just a shorthand for equivalence of types</p> </li> <li> <p>Strict judgmental equality of types could be thought of as making explicit the implicit <a class="existingWikiWord" href="/nlab/show/coercion">coercion</a> of <a class="existingWikiWord" href="/nlab/show/equivalence+of+types">equivalent types</a> as <a class="existingWikiWord" href="/nlab/show/subtypes">subtypes</a>, and is preserved throughout the type theory as <a class="existingWikiWord" href="/nlab/show/congruences">congruences</a>.</p> </li> </ul> <p>In either case, judgmental equality of types is primarily used for <a class="existingWikiWord" href="/nlab/show/definitional+equality">definitional equality</a> of types.</p> <h3 id="weak_judgmental_equality_2">Weak judgmental equality</h3> <p>Weak judgmental equality of types is given by one of the two sets of structural rules:</p> <ul> <li>The variable conversion rule for judgmentally equal types:<div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mo>≡</mo><mi>B</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>,</mo><mi>Δ</mi><mo>⊢</mo><mi>𝒥</mi></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>B</mi><mo>,</mo><mi>Δ</mi><mo>⊢</mo><mi>𝒥</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \equiv B \; \mathrm{type} \quad \Gamma, x:A, \Delta \vdash \mathcal{J}}{\Gamma, x:B, \Delta \vdash \mathcal{J}}</annotation></semantics></math></div></li> </ul> <p>or</p> <ul> <li>Rules for isomorphisms between judgmentally equal types:</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mo>≡</mo><mi>B</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><msub><mi>δ</mi> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><mi>B</mi></mrow></mfrac><mspace width="2em"></mspace><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mo>≡</mo><mi>B</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>y</mi><mo>:</mo><mi>B</mi><mo>⊢</mo><msubsup><mi>δ</mi> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><mi>A</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \equiv B \; \mathrm{type}}{\Gamma, x:A \vdash \delta_{A, B}(x):B} \qquad \frac{\Gamma \vdash A \equiv B \; \mathrm{type}}{\Gamma, y:B \vdash \delta_{A, B}^{-1}(x):A}</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mo>≡</mo><mi>B</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><msubsup><mi>δ</mi> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>δ</mi> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><mi>x</mi><mo>:</mo><mi>A</mi></mrow></mfrac><mspace width="2em"></mspace><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mo>≡</mo><mi>B</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>y</mi><mo>:</mo><mi>B</mi><mo>⊢</mo><msub><mi>δ</mi> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>δ</mi> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><mi>y</mi><mo>:</mo><mi>B</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \equiv B \; \mathrm{type}}{\Gamma, x:A \vdash \delta_{A, B}^{-1}(\delta_{A, B}(x)) \equiv x:A} \qquad \frac{\Gamma \vdash A \equiv B \; \mathrm{type}}{\Gamma, y:B \vdash \delta_{A, B}(\delta_{A, B}^{-1}(y)) \equiv y:B}</annotation></semantics></math></div> <p>In the first case, one could construct isomorphisms from the variable conversion rule, other structural rules, and the rules for function types:</p> <p>From the generic term rule and the variable conversion rule for judgmentally equal types <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>≡</mo><mi>A</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">A \equiv A'</annotation></semantics></math> we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>⊢</mo><mi>x</mi><mo>:</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">x:A' \vdash x:A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">x:A \vdash x:A'</annotation></semantics></math>, whereby from the introduction and computation rules for function types we have functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\lambda x:A'.x:A' \to A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>A</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\lambda x:A.x:A \to A'</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><mo stretchy="false">(</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>x</mi><mo>:</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">(\lambda x:A'.x)((\lambda x:A.x)(x)) \equiv (\lambda x:A'.x)(x) \equiv x:A</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><mo stretchy="false">(</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">(\lambda x:A.x)((\lambda x:A'.x)(x)) \equiv (\lambda x:A.x)(x) \equiv x:A'</annotation></semantics></math></div> <p>making both functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\lambda x:A'.x</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\lambda x:A.x</annotation></semantics></math> isomorphisms.</p> <h3 id="strict_judgmental_equality_2">Strict judgmental equality</h3> <p>In addition to the variable conversion rule, there are reflexivity, symmetry, and transitivity rules making strict judgmental equality for types an <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a>:</p> <ul> <li>Reflexivity of judgmental equality</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mo>≡</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash A \equiv A \; \mathrm{type}}</annotation></semantics></math></div> <ul> <li> <p>Symmetry of judgmental equality</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mo>≡</mo><mi>B</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><mi>B</mi><mo>≡</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \equiv B \; \mathrm{type}}{\Gamma \vdash B \equiv A \; \mathrm{type}}</annotation></semantics></math></div></li> <li> <p>Transitivity of judgmental equality</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mo>≡</mo><mi>B</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>B</mi><mo>≡</mo><mi>C</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mo>≡</mo><mi>C</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \equiv B \; \mathrm{type} \quad \Gamma \vdash B \equiv C \; \mathrm{type} }{\Gamma \vdash A \equiv C \; \mathrm{type}}</annotation></semantics></math></div></li> </ul> <h3 id="congruence_rules_for_judgmental_equality_of_types">Congruence rules for judgmental equality of types</h3> <p>In addition, strict judgmental equalities have <a class="existingWikiWord" href="/nlab/show/congruence+rules">congruence rules</a> for every type in the type theory.</p> <ul> <li>Congruence rules for dependent function types</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center"><mtr><mtd><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mo>≡</mo><mi>A</mi><mo>′</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mtd></mtr></mtable><mrow><mi>Γ</mi><mo>⊢</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≡</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo></mrow></munder><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{ \begin{array}{c} \Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \equiv B'(x) \; \mathrm{type} \end{array} }{\Gamma \vdash \prod_{x:A} B(x) \equiv \prod_{x:A'} B'(x)\; \mathrm{type}} </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center"><mtr><mtd><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>b</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>b</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>b</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>b</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr></mtable><mrow><mi>Γ</mi><mo>⊢</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>b</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>b</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</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><mo>.</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{ \begin{array}{c} \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash b(x):B(x) \quad \Gamma, x:A \vdash b'(x):B(x) \\ \Gamma, x:A \vdash b(x) \equiv b'(x):B(x) \end{array} }{\Gamma \vdash \lambda x:A.b(x) \equiv \lambda x:A.b'(x):\prod_{x:A}.B(x)} </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center"><mtr><mtd><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>f</mi><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="1em"></mspace><mi>f</mi><mo>′</mo><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mi>Γ</mi><mo>⊢</mo><mi>f</mi><mo>≡</mo><mi>f</mi><mo>′</mo><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr></mtable><mrow><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>f</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{ \begin{array}{c} \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma \vdash f:\prod_{x:A} B(x) \quad f':\prod_{x:A} B(x) \\ \Gamma \vdash f \equiv f':\prod_{x:A} B(x) \end{array} }{\Gamma, x:A \vdash f(x) \equiv f'(x):B(x)} </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center"><mtr><mtd><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>b</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>b</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>b</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>b</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr></mtable><mrow><mi>Γ</mi><mo>⊢</mo><msubsup><mi>β</mi> <mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msubsup><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>b</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≡</mo><msubsup><mi>β</mi> <mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msubsup><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>b</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</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><mi>b</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>b</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{ \begin{array}{c} \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash b(x):B(x) \quad \Gamma, x:A \vdash b'(x):B(x) \\ \Gamma, x:A \vdash b(x) \equiv b'(x):B(x) \end{array} }{\Gamma \vdash \beta_{\prod}^{A, B} x:A.b(x) \equiv \beta_{\prod}^{A, B} x:A.b'(x):\prod_{x:A} b(x) =_{B(x)} (\lambda x:A.b(x))(x)} </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center"><mtr><mtd><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mtd></mtr> <mtr><mtd><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mtd></mtr></mtable><mrow><mi>Γ</mi><mo>⊢</mo><msubsup><mi>η</mi> <mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msubsup><mo>≡</mo><msubsup><mi>η</mi> <mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>′</mo></mrow></msubsup><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>f</mi><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></munder><mi>f</mi><msub><mo>=</mo> <mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{ \begin{array}{c} \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash B'(x) \; \mathrm{type} \\ \Gamma, x:A \vdash B(x) \equiv B'(x) \; \mathrm{type} \end{array} }{\Gamma \vdash \eta_{\prod}^{A, B} \equiv \eta_{\prod}^{A, B'}:\prod_{f:\prod_{x:A} B(x)} f =_{\prod_{x:A} B(x)} \lambda x:A.f(x)} </annotation></semantics></math></div> <ul> <li>Congruence rules for dependent pair types:</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center"><mtr><mtd><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mo>≡</mo><mi>A</mi><mo>′</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mtd></mtr></mtable><mrow><mi>Γ</mi><mo>⊢</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≡</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo></mrow></munder><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{ \begin{array}{c} \Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \equiv B'(x) \; \mathrm{type} \end{array} }{\Gamma \vdash \sum_{x:A} B(x) \equiv \sum_{x:A'} B'(x)\; \mathrm{type}} </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center"><mtr><mtd><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mo>≡</mo><mi>A</mi><mo>′</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mtd></mtr></mtable><mrow><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>,</mo><mi>y</mi><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⊢</mo><msubsup><mi mathvariant="normal">pair</mi> <mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msubsup><mo>≡</mo><msubsup><mi mathvariant="normal">pair</mi> <mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>A</mi><mo>′</mo><mo>,</mo><mi>B</mi><mo>′</mo></mrow></msubsup><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{ \begin{array}{c} \Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \equiv B'(x) \; \mathrm{type} \end{array} }{\Gamma, x:A, y:B(x) \vdash \mathrm{pair}_{\sum}^{A, B} \equiv \mathrm{pair}_{\sum}^{A', B'}:\sum_{x:A} B(x)} </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center"><mtr><mtd><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mo>≡</mo><mi>A</mi><mo>′</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>z</mi><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⊢</mo><mi>C</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mtd></mtr></mtable><mrow><mi>Γ</mi><mo>⊢</mo><msubsup><mi mathvariant="normal">ind</mi> <mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi></mrow></msubsup><mo>≡</mo><msubsup><mi mathvariant="normal">ind</mi> <mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>A</mi><mo>′</mo><mo>,</mo><mi>B</mi><mo>′</mo><mo>,</mo><mi>C</mi><mo>′</mo></mrow></msubsup><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>g</mi><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>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></munder><mi>C</mi><mo stretchy="false">(</mo><msubsup><mi mathvariant="normal">pair</mi> <mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>z</mi><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></munder><mi>C</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{ \begin{array}{c} \Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \equiv B'(x) \; \mathrm{type} \quad \Gamma, z:\sum_{x:A} B(x) \vdash C(z) \equiv C'(z) \; \mathrm{type} \end{array} }{\Gamma \vdash \mathrm{ind}_{\sum}^{A, B, C} \equiv \mathrm{ind}_{\sum}^{A', B', C'}:\prod_{g:\prod_{x:A} \prod_{y:B(x)} C(\mathrm{pair}_{\sum}^{A, B}(x, y))} \prod_{z:\sum_{x:A} B(x)} C(z)} </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center"><mtr><mtd><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mo>≡</mo><mi>A</mi><mo>′</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>z</mi><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⊢</mo><mi>C</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mtd></mtr></mtable><mrow><mi>Γ</mi><mo>⊢</mo><msubsup><mi>β</mi> <mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi></mrow></msubsup><mo>≡</mo><msubsup><mi>β</mi> <mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>A</mi><mo>′</mo><mo>,</mo><mi>B</mi><mo>′</mo><mo>,</mo><mi>C</mi><mo>′</mo></mrow></msubsup><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>g</mi><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>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></munder><mi>C</mi><mo stretchy="false">(</mo><msubsup><mi mathvariant="normal">pair</mi> <mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></munder><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>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></munder><msubsup><mi mathvariant="normal">ind</mi> <mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><msubsup><mi mathvariant="normal">pair</mi> <mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>C</mi><mo stretchy="false">(</mo><msubsup><mi mathvariant="normal">pair</mi> <mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{ \begin{array}{c} \Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \equiv B'(x) \; \mathrm{type} \quad \Gamma, z:\sum_{x:A} B(x) \vdash C(z) \equiv C'(z) \; \mathrm{type} \end{array} }{\Gamma \vdash \beta_{\sum}^{A, B, C} \equiv \beta_{\sum}^{A', B', C'}:\prod_{g:\prod_{x:A} \prod_{y:B(x)} C(\mathrm{pair}_{\sum}^{A, B}(x, y))} \prod_{x:A} \prod_{y:B(x)} \mathrm{ind}_{\sum}^{A, B, C}(g, \mathrm{pair}_{\sum}^{A, B}(x, y)) =_{C(\mathrm{pair}_{\sum}^{A, B}(x, y))} g(x, y)} </annotation></semantics></math></div> <ul> <li>Congruence rules for identity types:</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mo>≡</mo><mi>A</mi><mo>′</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>,</mo><mi>y</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>x</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>y</mi><mo>≡</mo><mi>x</mi><msub><mo>=</mo> <mrow><mi>A</mi><mo>′</mo></mrow></msub><mi>y</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \equiv A' \; \mathrm{type}}{\Gamma, x:A, y:A \vdash x =_A y \equiv x =_{A'} y}</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mo>≡</mo><mi>A</mi><mo>′</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><msub><mi mathvariant="normal">refl</mi> <mi>A</mi></msub><mo>≡</mo><msub><mi mathvariant="normal">refl</mi> <mrow><mi>A</mi><mo>′</mo></mrow></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><mi>x</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \equiv A' \; \mathrm{type}}{\Gamma \vdash \mathrm{refl}_A \equiv \mathrm{refl}_{A'}:\prod_{x:A} x =_A x}</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center"><mtr><mtd><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mo>≡</mo><mi>A</mi><mo>′</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>,</mo><mi>y</mi><mo>:</mo><mi>A</mi><mo>,</mo><mi>p</mi><mo>:</mo><mi>x</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>y</mi><mo>⊢</mo><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mtd></mtr></mtable><mrow><mi>Γ</mi><mo>⊢</mo><msubsup><mi mathvariant="normal">ind</mi> <mo>=</mo> <mrow><mi>A</mi><mo>,</mo><mi>C</mi></mrow></msubsup><mo>≡</mo><msubsup><mi mathvariant="normal">ind</mi> <mo>=</mo> <mrow><mi>A</mi><mo>′</mo><mo>,</mo><mi>C</mi><mo>′</mo></mrow></msubsup><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>t</mi><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">refl</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></munder><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><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>p</mi><mo>:</mo><mi>x</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>y</mi></mrow></munder><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{ \begin{array}{c} \Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A, y:A, p:x =_A y \vdash C(x, y, p) \equiv C'(x, y, p) \; \mathrm{type} \end{array} }{\Gamma \vdash \mathrm{ind}_{=}^{A, C} \equiv \mathrm{ind}_{=}^{A', C'}:\prod_{t:\prod_{x:A} C(x, x, \mathrm{refl}_A(x))} \prod_{x:A} \prod_{y:A} \prod_{p:x =_A y} C(x, y, p)} </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center"><mtr><mtd><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mo>≡</mo><mi>A</mi><mo>′</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>,</mo><mi>y</mi><mo>:</mo><mi>A</mi><mo>,</mo><mi>p</mi><mo>:</mo><mi>x</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>y</mi><mo>⊢</mo><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mtd></mtr></mtable><mrow><mi>Γ</mi><mo>⊢</mo><msubsup><mi>β</mi> <mrow><msub><mi mathvariant="normal">ind</mi> <mo>=</mo></msub></mrow> <mrow><mi>A</mi><mo>,</mo><mi>C</mi></mrow></msubsup><mo>≡</mo><msubsup><mi>β</mi> <mrow><msub><mi mathvariant="normal">ind</mi> <mo>=</mo></msub></mrow> <mrow><mi>A</mi><mo>′</mo><mo>,</mo><mi>C</mi><mo>′</mo></mrow></msubsup><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>t</mi><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">refl</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><msubsup><mi mathvariant="normal">ind</mi> <mo>=</mo> <mrow><mi>A</mi><mo>,</mo><mi>C</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">refl</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">refl</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mi>t</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{ \begin{array}{c} \Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A, y:A, p:x =_A y \vdash C(x, y, p) \equiv C'(x, y, p) \; \mathrm{type} \end{array} }{\Gamma \vdash \beta_{ \mathrm{ind}_=}^{A, C} \equiv \beta_{\mathrm{ind}_=}^{A', C'}:\prod_{t:\prod_{x:A} C(x, x, \mathrm{refl}_A(x))} \prod_{x:A} \mathrm{ind}_{=}^{A, C}(t, x, x, \mathrm{refl}_A(x)) =_{C(x, x, \mathrm{refl}_A(x))} t(x)} </annotation></semantics></math></div> <ul> <li>Congruence rules for the empty type:</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>∅</mi><mo>⊢</mo><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><msubsup><mi mathvariant="normal">ind</mi> <mi>∅</mi> <mi>C</mi></msubsup><mo>≡</mo><msubsup><mi mathvariant="normal">ind</mi> <mi>∅</mi> <mrow><mi>C</mi><mo>′</mo></mrow></msubsup><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>∅</mi></mrow></munder><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma, x:\emptyset \vdash C(x) \equiv C'(x) \; \mathrm{type}}{\Gamma \vdash \mathrm{ind}_\emptyset^C \equiv \mathrm{ind}_\emptyset^{C'}:\prod_{x:\emptyset} C(x) \; \mathrm{type}}</annotation></semantics></math></div> <ul> <li>Congruence rules for the type of booleans:</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>𝟚</mi><mo>⊢</mo><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><msubsup><mi mathvariant="normal">ind</mi> <mi>𝟚</mi> <mi>C</mi></msubsup><mo>≡</mo><msubsup><mi mathvariant="normal">ind</mi> <mi>𝟚</mi> <mrow><mi>C</mi><mo>′</mo></mrow></msubsup><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>a</mi><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>b</mi><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>𝟚</mi></mrow></munder><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma, x:\mathbb{2} \vdash C(x) \equiv C'(x) \; \mathrm{type}}{\Gamma \vdash \mathrm{ind}_\mathbb{2}^C \equiv \mathrm{ind}_\mathbb{2}^{C'}:\prod_{a:C(0)} \prod_{b:C(1)} \prod_{x:\mathbb{2}} C(x)}</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>𝟚</mi><mo>⊢</mo><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><msubsup><mi>β</mi> <mi>𝟚</mi> <mrow><mn>0</mn><mo>,</mo><mi>C</mi></mrow></msubsup><mo>≡</mo><msubsup><mi>β</mi> <mi>𝟚</mi> <mrow><mn>0</mn><mo>,</mo><mi>C</mi><mo>′</mo></mrow></msubsup><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>a</mi><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>b</mi><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></munder><msubsup><mi mathvariant="normal">ind</mi> <mi>𝟚</mi> <mi>C</mi></msubsup><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>C</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msub><mi>a</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma, x:\mathbb{2} \vdash C(x) \equiv C'(x) \; \mathrm{type}}{\Gamma \vdash \beta_\mathbb{2}^{0, C} \equiv \beta_\mathbb{2}^{0, C'}:\prod_{a:C(0)} \prod_{b:C(1)} \mathrm{ind}_\mathbb{2}^C(a, b, 0) =_{C(0)} a}</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>𝟚</mi><mo>⊢</mo><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><msubsup><mi>β</mi> <mi>𝟚</mi> <mrow><mn>1</mn><mo>,</mo><mi>C</mi></mrow></msubsup><mo>≡</mo><msubsup><mi>β</mi> <mi>𝟚</mi> <mrow><mn>1</mn><mo>,</mo><mi>C</mi><mo>′</mo></mrow></msubsup><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>a</mi><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>b</mi><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></munder><msubsup><mi mathvariant="normal">ind</mi> <mi>𝟚</mi> <mi>C</mi></msubsup><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>C</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mi>b</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma, x:\mathbb{2} \vdash C(x) \equiv C'(x) \; \mathrm{type}}{\Gamma \vdash \beta_\mathbb{2}^{1, C} \equiv \beta_\mathbb{2}^{1, C'}:\prod_{a:C(0)} \prod_{b:C(1)} \mathrm{ind}_\mathbb{2}^C(a, b, 1) =_{C(1)} b}</annotation></semantics></math></div> <ul> <li>Congruence rules for the natural numbers type:</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>ℕ</mi><mo>⊢</mo><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><msubsup><mi mathvariant="normal">ind</mi> <mi>ℕ</mi> <mi>C</mi></msubsup><mo>≡</mo><msubsup><mi mathvariant="normal">ind</mi> <mi>ℕ</mi> <mrow><mi>C</mi><mo>′</mo></mrow></msubsup><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><msub><mi>c</mi> <mn>0</mn></msub><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><msub><mi>c</mi> <mi>s</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>ℕ</mi><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></munder></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma, x:\mathbb{N} \vdash C(x) \equiv C'(x) \; \mathrm{type}}{\Gamma \vdash \mathrm{ind}_\mathbb{N}^C \equiv \mathrm{ind}_\mathbb{N}^{C'}:\prod_{c_0:C(0)} \prod_{c_s:\prod_{x:\mathbb{N}} C(x) \to C(s(x))} \prod_{x:\mathbb{N} C(x)}}</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>ℕ</mi><mo>⊢</mo><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><msubsup><mi>β</mi> <mi>ℕ</mi> <mrow><mn>0</mn><mo>,</mo><mi>C</mi></mrow></msubsup><mo>≡</mo><msubsup><mi>β</mi> <mi>ℕ</mi> <mrow><mn>0</mn><mo>,</mo><mi>C</mi><mo>′</mo></mrow></msubsup><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><msub><mi>c</mi> <mn>0</mn></msub><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><msub><mi>c</mi> <mi>s</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></munder><msubsup><mi mathvariant="normal">ind</mi> <mi>ℕ</mi> <mi>C</mi></msubsup><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>C</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msub><msub><mi>c</mi> <mn>0</mn></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma, x:\mathbb{N} \vdash C(x) \equiv C'(x) \; \mathrm{type}}{\Gamma \vdash \beta_\mathbb{N}^{0, C} \equiv \beta_\mathbb{N}^{0, C'}:\prod_{c_0:C(0)} \prod_{c_s:\prod_{x:\mathbb{N}} C(x) \to C(s(x))} \mathrm{ind}_\mathbb{N}^C(c_0, c_s, 0) =_{C(0)} c_0}</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>ℕ</mi><mo>⊢</mo><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><msubsup><mi>β</mi> <mi>ℕ</mi> <mrow><mi>s</mi><mo>,</mo><mi>C</mi></mrow></msubsup><mo>≡</mo><msubsup><mi>β</mi> <mi>ℕ</mi> <mrow><mi>s</mi><mo>,</mo><mi>C</mi><mo>′</mo></mrow></msubsup><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><msub><mi>c</mi> <mn>0</mn></msub><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><msub><mi>c</mi> <mi>s</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><msubsup><mi mathvariant="normal">ind</mi> <mi>ℕ</mi> <mi>C</mi></msubsup><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><mi>s</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>C</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><msub><mi>c</mi> <mi>s</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msubsup><mi mathvariant="normal">ind</mi> <mi>ℕ</mi> <mi>C</mi></msubsup><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma, x:\mathbb{N} \vdash C(x) \equiv C'(x) \; \mathrm{type}}{\Gamma \vdash \beta_\mathbb{N}^{s, C} \equiv \beta_\mathbb{N}^{s, C'}:\prod_{c_0:C(0)} \prod_{c_s:\prod_{x:\mathbb{N}} C(x) \to C(s(x))} \prod_{x:\mathbb{N}} \mathrm{ind}_\mathbb{N}^C(c_0, c_s, s(x)) =_{C(s(x))} c_s(x)(\mathrm{ind}_\mathbb{N}^C(c_0, c_s, x))}</annotation></semantics></math></div> <p>Similarly, we have congruence rules for every <a class="existingWikiWord" href="/nlab/show/axiom">axiom</a> in the dependent type theory, such as</p> <ul> <li>congruence rule for <a class="existingWikiWord" href="/nlab/show/function+extensionality">function extensionality</a>:</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mo>≡</mo><mi>A</mi><mo>′</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><msub><mi mathvariant="normal">funext</mi> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msub><mo>≡</mo><msub><mi mathvariant="normal">funext</mi> <mrow><mi>A</mi><mo>′</mo><mo>,</mo><mi>B</mi><mo>′</mo></mrow></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>f</mi><mo>;</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>g</mi><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></munder><mo stretchy="false">(</mo><mi>f</mi><msub><mo>=</mo> <mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mi>g</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><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \equiv B'(x) \; \mathrm{type}}{\Gamma \vdash \mathrm{funext}_{A, B} \equiv \mathrm{funext}_{A', B'}:\prod_{f;\prod_{x:A} B(x)} \prod_{g:\prod_{x:A} B(x)} (f =_{\prod_{x:A} B(x)} g) \simeq \prod_{x:A} f(x) =_{B(x)} g(x)}</annotation></semantics></math></div> <ul> <li>congruence rule for the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a>:</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mo>≡</mo><mi>A</mi><mo>′</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>,</mo><mi>y</mi><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⊢</mo><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><msub><mi mathvariant="normal">choice</mi> <mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi></mrow></msub><mo>≡</mo><msub><mi mathvariant="normal">choice</mi> <mrow><mi>A</mi><mo>′</mo><mo>,</mo><mi>B</mi><mo>′</mo><mo>,</mo><mi>C</mi><mo>′</mo></mrow></msub><mo>:</mo><mrow><mo>(</mo><mi mathvariant="normal">isSet</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><mi mathvariant="normal">isSet</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>→</mo><mo>∀</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><mo>∃</mo><mi>y</mi><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>.</mo><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>→</mo><mo>∃</mo><mi>g</mi><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>.</mo><mo>∀</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \equiv B'(x) \; \mathrm{type} \quad \Gamma, x:A, y:B(x) \vdash C(x, y) \equiv C'(x, y) \; \mathrm{type}}{\Gamma \vdash \mathrm{choice}_{A, B, C} \equiv \mathrm{choice}_{A', B', C'}:\left(\mathrm{isSet}(A) \times \prod_{x:A} \mathrm{isSet}(B(x))\right) \to \forall x:A.\exists y:B(x).C(x, y) \to \exists g:\prod_{x:A} B(x).\forall x:A.C(x, g(x))}</annotation></semantics></math></div> <h2 id="judgmental_equality_of_contexts">Judgmental equality of contexts</h2> <p>In some <a class="existingWikiWord" href="/nlab/show/dependent+type+theories">dependent type theories</a>, there is also judgmental equality of <a class="existingWikiWord" href="/nlab/show/contexts">contexts</a>, which is given by the following <a class="existingWikiWord" href="/nlab/show/judgment">judgment</a>:</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo>≡</mo><mi>Γ</mi><mo>′</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">ctx</mi></mrow><annotation encoding="application/x-tex">\Gamma \equiv \Gamma' \; \mathrm{ctx}</annotation></semantics></math> - <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\Gamma'</annotation></semantics></math> are judgmentally equal contexts.</li> </ul> <p>In addition to the variable conversion rule, there are reflexivity, symmetry, and transitivity rules making judgmental equality for contexts an <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a>:</p> <ul> <li>Reflexivity of judgmental equality</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">ctx</mi></mrow><mrow><mi>Γ</mi><mo>≡</mo><mi>Γ</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">ctx</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \; \mathrm{ctx}}{\Gamma \equiv \Gamma \; \mathrm{ctx}}</annotation></semantics></math></div> <ul> <li> <p>Symmetry of judgmental equality</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>≡</mo><mi>Δ</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">ctx</mi></mrow><mrow><mi>Δ</mi><mo>≡</mo><mi>Γ</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">ctx</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \equiv \Delta \; \mathrm{ctx}}{\Delta \equiv \Gamma \; \mathrm{ctx}}</annotation></semantics></math></div></li> <li> <p>Transitivity of judgmental equality</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>≡</mo><mi>Δ</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">ctx</mi><mspace width="1em"></mspace><mi>Δ</mi><mo>≡</mo><mi>Ξ</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">ctx</mi></mrow><mrow><mi>Γ</mi><mo>≡</mo><mi>Ξ</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">ctx</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \equiv \Delta \; \mathrm{ctx} \quad \Delta \equiv \Xi \; \mathrm{ctx}}{\Gamma \equiv \Xi \; \mathrm{ctx}}</annotation></semantics></math></div></li> </ul> <h2 id="see_also">See also</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equality">equality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/propositional+equality">propositional equality</a>, <a class="existingWikiWord" href="/nlab/show/typal+equality">typal equality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coercion">coercion</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li> <p>Robin Adams, <em>Pure type systems with judgemental equality</em>, Journal of Functional Programming, Volume 16 Issue 2(2006) (<a href="http://dl.acm.org/citation.cfm?id=1114675">web</a>)</p> </li> <li> <p>Vincent Siles, Hugo Herbelin, <em>Equality is typable in semi-full pure type systems</em> (<a href="http://pauillac.inria.fr/~herbelin/publis/lics-SilHer10-pts-typed-conv.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Egbert+Rijke">Egbert Rijke</a>, <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Type+Theory">Introduction to Homotopy Type Theory</a></em>, Cambridge Studies in Advanced Mathematics, Cambridge University Press (<a href="https://raw.githubusercontent.com/martinescardo/HoTTEST-Summer-School/main/HoTT/hott-intro.pdf">pdf</a>) (478 pages)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 17, 2024 at 12:40:03. 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