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href="/nlab/show/judgement">judgement</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypothetical+judgement">hypothetical judgement</a>, <a class="existingWikiWord" href="/nlab/show/sequent">sequent</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/antecedents">antecedents</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊢</mo></mrow><annotation encoding="application/x-tex">\vdash</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/consequent">consequent</a>, <a class="existingWikiWord" href="/nlab/show/succedents">succedents</a></li> </ul> </li> </ul> <ol> <li><a class="existingWikiWord" href="/nlab/show/type+formation+rule">type formation rule</a></li> <li><a class="existingWikiWord" href="/nlab/show/term+introduction+rule">term introduction rule</a></li> <li><a class="existingWikiWord" href="/nlab/show/term+elimination+rule">term elimination rule</a></li> <li><a class="existingWikiWord" href="/nlab/show/computation+rule">computation rule</a></li> </ol> <p><strong><a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a></strong> (<a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent</a>, <a class="existingWikiWord" href="/nlab/show/intensional+type+theory">intensional</a>, <a class="existingWikiWord" href="/nlab/show/observational+type+theory">observational type theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>)</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/calculus+of+constructions">calculus of constructions</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/syntax">syntax</a></strong> <a class="existingWikiWord" href="/nlab/show/object+language">object language</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/theory">theory</a>, <a class="existingWikiWord" href="/nlab/show/axiom">axiom</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proposition">proposition</a>/<a class="existingWikiWord" href="/nlab/show/type">type</a> (<a class="existingWikiWord" href="/nlab/show/propositions+as+types">propositions as types</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/definition">definition</a>/<a class="existingWikiWord" href="/nlab/show/proof">proof</a>/<a class="existingWikiWord" href="/nlab/show/program">program</a> (<a class="existingWikiWord" href="/nlab/show/proofs+as+programs">proofs as programs</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/theorem">theorem</a></p> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/computational+trinitarianism">computational trinitarianism</a></strong> = <br /> <strong><a class="existingWikiWord" href="/nlab/show/propositions+as+types">propositions as types</a></strong> +<strong><a class="existingWikiWord" href="/nlab/show/programs+as+proofs">programs as proofs</a></strong> +<strong><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation type theory/category theory</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/logic">logic</a></th><th><a class="existingWikiWord" href="/nlab/show/set+theory">set theory</a> (<a class="existingWikiWord" href="/nlab/show/internal+logic+of+set+theory">internal logic</a> of)</th><th><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></th><th><a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/proposition">proposition</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/set">set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/object">object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type">type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/predicate">predicate</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/family+of+sets">family of sets</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/display+morphism">display morphism</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+type">dependent type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/proof">proof</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/element">element</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/generalized+element">generalized element</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/term">term</a>/<a class="existingWikiWord" href="/nlab/show/program">program</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cut+rule">cut rule</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/composition">composition</a> of <a class="existingWikiWord" href="/nlab/show/classifying+morphisms">classifying morphisms</a> / <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of <a class="existingWikiWord" href="/nlab/show/display+maps">display maps</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/substitution">substitution</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/introduction+rule">introduction rule</a> for <a class="existingWikiWord" href="/nlab/show/implication">implication</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/counit">counit</a> for hom-tensor adjunction</td><td style="text-align: left;">lambda</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/elimination+rule">elimination rule</a> for <a class="existingWikiWord" href="/nlab/show/implication">implication</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/unit">unit</a> for hom-tensor adjunction</td><td style="text-align: left;">application</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cut+elimination">cut elimination</a> for <a class="existingWikiWord" href="/nlab/show/implication">implication</a></td><td style="text-align: left;"></td><td style="text-align: left;">one of the <a class="existingWikiWord" href="/nlab/show/zigzag+identities">zigzag identities</a> for hom-tensor adjunction</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/beta+reduction">beta reduction</a></td></tr> <tr><td style="text-align: left;">identity elimination for <a class="existingWikiWord" href="/nlab/show/implication">implication</a></td><td style="text-align: left;"></td><td style="text-align: left;">the other <a class="existingWikiWord" href="/nlab/show/zigzag+identity">zigzag identity</a> for hom-tensor adjunction</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/eta+conversion">eta conversion</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/true">true</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/singleton">singleton</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>/<a class="existingWikiWord" href="/nlab/show/%28-2%29-truncated+object">(-2)-truncated object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/h-level+0">h-level 0</a>-<a class="existingWikiWord" href="/nlab/show/type">type</a>/<a class="existingWikiWord" href="/nlab/show/unit+type">unit type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/false">false</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/empty+set">empty set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/empty+type">empty type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/proposition">proposition</a>, <a class="existingWikiWord" href="/nlab/show/truth+value">truth value</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/subsingleton">subsingleton</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/subterminal+object">subterminal object</a>/<a class="existingWikiWord" href="/nlab/show/%28-1%29-truncated+object">(-1)-truncated object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/h-proposition">h-proposition</a>, <a class="existingWikiWord" href="/nlab/show/mere+proposition">mere proposition</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/logical+conjunction">logical conjunction</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cartesian+product">cartesian product</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/product">product</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/product+type">product type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/disjunction">disjunction</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a> (<a class="existingWikiWord" href="/nlab/show/support">support</a> of)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> (<a class="existingWikiWord" href="/nlab/show/%28-1%29-truncation">(-1)-truncation</a> of)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/sum+type">sum type</a> (<a class="existingWikiWord" href="/nlab/show/bracket+type">bracket type</a> of)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/implication">implication</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/function+set">function set</a> (into <a class="existingWikiWord" href="/nlab/show/subsingleton">subsingleton</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> (into <a class="existingWikiWord" href="/nlab/show/subterminal+object">subterminal object</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/function+type">function type</a> (into <a class="existingWikiWord" href="/nlab/show/h-proposition">h-proposition</a>)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/negation">negation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/function+set">function set</a> into <a class="existingWikiWord" href="/nlab/show/empty+set">empty set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> into <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/function+type">function type</a> into <a class="existingWikiWord" href="/nlab/show/empty+type">empty type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/universal+quantification">universal quantification</a></td><td style="text-align: left;">indexed <a class="existingWikiWord" href="/nlab/show/cartesian+product">cartesian product</a> (of family of <a class="existingWikiWord" href="/nlab/show/subsingletons">subsingletons</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a> (of family of <a class="existingWikiWord" href="/nlab/show/subterminal+objects">subterminal objects</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+product+type">dependent product type</a> (of family of <a class="existingWikiWord" href="/nlab/show/h-propositions">h-propositions</a>)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/existential+quantification">existential quantification</a></td><td style="text-align: left;">indexed <a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a> (<a class="existingWikiWord" href="/nlab/show/support">support</a> of)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a> (<a class="existingWikiWord" href="/nlab/show/%28-1%29-truncation">(-1)-truncation</a> of)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+sum+type">dependent sum type</a> (<a class="existingWikiWord" href="/nlab/show/bracket+type">bracket type</a> of)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/logical+equivalence">logical equivalence</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/bijection+set">bijection set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/object+of+isomorphisms">object of isomorphisms</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/equivalence+type">equivalence type</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/support+set">support set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/support+object">support object</a>/<a class="existingWikiWord" href="/nlab/show/%28-1%29-truncation">(-1)-truncation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/propositional+truncation">propositional truncation</a>/<a class="existingWikiWord" href="/nlab/show/bracket+type">bracket type</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/n-image">n-image</a> of <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> into <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>/<a class="existingWikiWord" href="/nlab/show/n-truncation">n-truncation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/n-truncation+modality">n-truncation modality</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/equality">equality</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/diagonal+function">diagonal function</a>/<a class="existingWikiWord" href="/nlab/show/diagonal+subset">diagonal subset</a>/<a class="existingWikiWord" href="/nlab/show/diagonal+relation">diagonal relation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/identity+type">identity type</a>/<a class="existingWikiWord" href="/nlab/show/path+type">path type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/completely+presented+set">completely presented set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/set">set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/discrete+object">discrete object</a>/<a class="existingWikiWord" href="/nlab/show/0-truncated+object">0-truncated object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/h-level+2">h-level 2</a>-<a class="existingWikiWord" href="/nlab/show/type">type</a>/<a class="existingWikiWord" href="/nlab/show/set">set</a>/<a class="existingWikiWord" href="/nlab/show/h-set">h-set</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/set">set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/set">set</a> with <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28infinity%2C1%29-category">internal 0-groupoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Bishop+set">Bishop set</a>/<a class="existingWikiWord" href="/nlab/show/setoid">setoid</a> with its <a class="existingWikiWord" href="/nlab/show/pseudo-equivalence+relation">pseudo-equivalence relation</a> an actual <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/equivalence+class">equivalence class</a>/<a class="existingWikiWord" href="/nlab/show/quotient+set">quotient set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quotient">quotient</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quotient+type">quotient type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/induction">induction</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/inductive+type">inductive type</a>, <a class="existingWikiWord" href="/nlab/show/W-type">W-type</a>, <a class="existingWikiWord" href="/nlab/show/M-type">M-type</a></td></tr> <tr><td style="text-align: left;">higher <a class="existingWikiWord" href="/nlab/show/induction">induction</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-colimit">higher colimit</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/higher+inductive+type">higher inductive type</a></td></tr> <tr><td style="text-align: left;">-</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/0-truncated">0-truncated</a> <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-colimit">higher colimit</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quotient+inductive+type">quotient inductive type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coinduction">coinduction</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/limit">limit</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coinductive+type">coinductive type</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/preset">preset</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type">type</a> without <a class="existingWikiWord" href="/nlab/show/identity+types">identity types</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/set">set</a> of <a class="existingWikiWord" href="/nlab/show/truth+values">truth values</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+of+propositions">type of propositions</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/domain+of+discourse">domain of discourse</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/universe">universe</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/object+classifier">object classifier</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+universe">type universe</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/modality">modality</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/closure+operator">closure operator</a>, (<a class="existingWikiWord" href="/nlab/show/idempotent+monad">idempotent</a>) <a class="existingWikiWord" href="/nlab/show/monad">monad</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/modal+type+theory">modal type theory</a>, <a class="existingWikiWord" href="/nlab/show/monad+%28in+computer+science%29">monad (in computer science)</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a></td><td style="text-align: left;"></td><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric</a>, <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed</a>) <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+type+theory">linear type theory</a>/<a class="existingWikiWord" href="/nlab/show/quantum+computation">quantum computation</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/proof+net">proof net</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/string+diagram">string diagram</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quantum+circuit">quantum circuit</a></td></tr> <tr><td style="text-align: left;">(absence of) <a class="existingWikiWord" href="/nlab/show/contraction+rule">contraction rule</a></td><td style="text-align: left;"></td><td style="text-align: left;">(absence of) <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/no-cloning+theorem">no-cloning theorem</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/synthetic+mathematics">synthetic mathematics</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/domain+specific+embedded+programming+language">domain specific embedded programming language</a></td></tr> </tbody></table> </div> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+levels">homotopy levels</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-type+theory">2-type theory</a>, <a class="existingWikiWord" href="/michaelshulman/show/2-categorical+logic">2-categorical logic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory+-+contents">homotopy type theory - contents</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/univalence">univalence</a>, <a class="existingWikiWord" href="/nlab/show/function+extensionality">function extensionality</a>, <a class="existingWikiWord" href="/nlab/show/internal+logic+of+an+%28%E2%88%9E%2C1%29-topos">internal logic of an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+type+theory">cohesive homotopy type theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/directed+homotopy+type+theory">directed homotopy type theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/HoTT+methods+for+homotopy+theorists">HoTT methods for homotopy theorists</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/semantics">semantics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a>, <a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/display+map">display map</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic+of+a+topos">internal logic of a topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Mitchell-Benabou+language">Mitchell-Benabou language</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kripke-Joyal+semantics">Kripke-Joyal semantics</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic+of+an+%28%E2%88%9E%2C1%29-topos">internal logic of an (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/type-theoretic+model+category">type-theoretic model category</a></li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/type+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="dependent_product_types">Dependent product types</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#overview'>Overview</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#as_a_negative_type'>As a negative type</a></li> <li><a href='#as_a_positive_type'>As a positive type</a></li> <li><a href='#positive_versus_negative'>Positive versus negative</a></li> <li><a href='#dependent_product_types_a_la_russell_and_a_la_tarski'>Dependent product types a la Russell and a la Tarski</a></li> <li><a href='#weak_and_strict_dependent_product_types'>Weak and strict dependent product types</a></li> <li><a href='#in_terms_of_function_types'>In terms of function types</a></li> <li><a href='#as_types_of_dependent_anafunctions'>As types of dependent anafunctions</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#universal_property_of_dependent_product_types'>Universal property of dependent product types</a></li> <li><a href='#typal_computation_and_uniqueness_rules'>Typal computation and uniqueness rules</a></li> <li><a href='#TypalCongruenceRules'>Typal congruence rules</a></li> <ul> <li><a href='#strict_dependent_product_types'>Strict dependent product types</a></li> <ul> <li><a href='#using_definitional_isomorphisms'>Using definitional isomorphisms</a></li> <li><a href='#using_weak_equivalences_of_types'>Using weak equivalences of types</a></li> <li><a href='#using_identity_types_between_types'>Using identity types between types</a></li> </ul> <li><a href='#weak_dependent_product_types'>Weak dependent product types</a></li> </ul> <li><a href='#application_in_logic'>Application in logic</a></li> <li><a href='#graph_of_a_dependent_function'>Graph of a dependent function</a></li> <li><a href='#relation_to_sections'>Relation to sections</a></li> </ul> <li><a href='#categorical_interpretation'>Categorical interpretation</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>In <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a>, a <em>dependent product type</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\prod_{x\colon A} B(x)</annotation></semantics></math>, for a <a class="existingWikiWord" href="/nlab/show/dependent+type">dependent type</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>⊢</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>Type</mi></mrow><annotation encoding="application/x-tex">x\colon A\vdash B(x)\colon Type</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/type">type</a> of “dependently typed <a class="existingWikiWord" href="/nlab/show/functions">functions</a>” assigning to each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">x\colon A</annotation></semantics></math> an element of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(x)</annotation></semantics></math>.</p> <p>In a <a class="existingWikiWord" href="/nlab/show/model">model</a> of the type theory in <a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a>, this is a <a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a>. In <a class="existingWikiWord" href="/nlab/show/set+theory">set theory</a>, it is an element of an indexed product.</p> <p>It includes <a class="existingWikiWord" href="/nlab/show/function+types">function types</a> as the special case when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is not dependent on <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>, <a class="existingWikiWord" href="/nlab/show/product+types">product types</a> as a special case when <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> is the type of <a class="existingWikiWord" href="/nlab/show/Booleans">Booleans</a>, and <a class="existingWikiWord" href="/nlab/show/dependent+sequence+types">dependent sequence types</a> as a special case when <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> is the <a class="existingWikiWord" href="/nlab/show/natural+numbers+type">natural numbers type</a>.</p> <h2 id="overview">Overview</h2> <div> <p>The <a class="existingWikiWord" href="/nlab/show/inference+rules">inference rules</a> for <a class="existingWikiWord" href="/nlab/show/dependent+function+types">dependent function types</a> (aka “dependent product types” or “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math>-types”):</p> <div style="margin: -30px 0px 20px 10px"> <img src="/nlab/files/DependentFunctionTypeInference-230215.jpg" width="770px" /> </div></div> <h2 id="definition">Definition</h2> <p>Like any type constructor in type theory, a dependent product type is specified by rules saying when we can introduce it as a type, how to construct terms of that type, how to use or “eliminate” terms of that type, and how to compute when we combine the constructors with the eliminators.</p> <p>The <a class="existingWikiWord" href="/nlab/show/type+formation+rule">type formation rule</a> for dependent product type is:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>A</mi><mo lspace="verythinmathspace">:</mo><mi>Type</mi><mspace width="2em"></mspace><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>⊢</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>Type</mi></mrow><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi></mrow></munder><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>Type</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{A\colon Type \qquad x\colon A \vdash B(x) \colon Type}{\prod_{x\colon A} B(x)\colon Type}</annotation></semantics></math></div> <h3 id="as_a_negative_type">As a negative type</h3> <p>Dependent product types are almost always defined as <a class="existingWikiWord" href="/nlab/show/negative+types">negative types</a>. In this presentation, primacy is given to the eliminators. The natural eliminator of a dependent product type says that we can <em>apply</em> it to any input:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi></mrow></munder><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="2em"></mspace><mi>a</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi></mrow><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{f\colon \prod_{x\colon A} B(x) \qquad a\colon A}{f(a) \colon B(a)}</annotation></semantics></math></div> <p>The constructor is then determined as usual for a negative type: to construct a term of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\prod_{x\colon A} B(x)</annotation></semantics></math>, we have to specify how it behaves when applied to any input. In other words, we should have a term of type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(x)</annotation></semantics></math> containing a free variable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">x\colon A</annotation></semantics></math>. This yields the usual “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math>-abstraction” constructor:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>⊢</mo><mi>b</mi><mo lspace="verythinmathspace">:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>λ</mi><mi>x</mi><mo>.</mo><mi>b</mi><mo lspace="verythinmathspace">:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo lspace="verythinmathspace">:</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{x\colon A\vdash b\colon B(x)}{\lambda x.b\colon \prod_{x\colon A} B(x)}</annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/beta-reduction">beta-reduction</a> rule is the obvious one, saying that when we evaluate a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math>-abstraction, we do it by substituting for the bound variable.</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>b</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><msub><mo>→</mo> <mi>β</mi></msub><mspace width="thickmathspace"></mspace><mi>b</mi><mo stretchy="false">[</mo><mi>a</mi><mo stretchy="false">/</mo><mi>x</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">(\lambda x.b)(a) \;\to_\beta\; b[a/x]</annotation></semantics></math></div> <p>If we want an <a class="existingWikiWord" href="/nlab/show/eta-conversion">eta-conversion</a> rule, the natural one says that every dependently typed function is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math>-abstraction:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mi>x</mi><mo>.</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><msub><mo>→</mo> <mi>η</mi></msub><mspace width="thickmathspace"></mspace><mi>f</mi></mrow><annotation encoding="application/x-tex"> \lambda x.f(x) \;\to_\eta\; f</annotation></semantics></math></div> <h3 id="as_a_positive_type">As a positive type</h3> <p>It is also possible to present dependent product types as a <a class="existingWikiWord" href="/nlab/show/positive+type">positive type</a>. However, this requires a stronger metatheory, such as a <a class="existingWikiWord" href="/nlab/show/logical+framework">logical framework</a>. We use the same constructor (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math>-abstraction), but now the eliminator says that to define an operation using a function, it suffices to say what to do in the case that that function is a lambda abstraction.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mo stretchy="false">(</mo><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>⊢</mo><mi>b</mi><mo lspace="verythinmathspace">:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⊢</mo><mi>c</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mspace width="2em"></mspace><mi>f</mi><mo lspace="verythinmathspace">:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi></mrow></munder><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>funsplit</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>C</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{(x\colon A \vdash b\colon B(x)) \vdash c\colon C \qquad f\colon \prod_{x\colon A} B(x)}{funsplit(c,f)\colon C}</annotation></semantics></math></div> <p>This rule cannot be formulated in the usual presentation of type theory, since it involves a “higher-order judgment”: the context of the term <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c\colon C</annotation></semantics></math> involves a “term of type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(x)</annotation></semantics></math> containing a free variable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">x\colon A</annotation></semantics></math>”. However, it is possible to make sense of it. In <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a>, we need additionally to allow <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> to depend on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\prod_{x\colon A} B(x)</annotation></semantics></math>.</p> <p>The natural <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math>-reduction rule for this eliminator is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>funsplit</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>λ</mi><mi>x</mi><mo>.</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><msub><mo>→</mo> <mi>β</mi></msub><mi>c</mi><mo stretchy="false">[</mo><mi>g</mi><mo stretchy="false">/</mo><mi>b</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> funsplit(c, \lambda x.g) \;\to_\beta c[g/b] </annotation></semantics></math></div> <p>and its <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math>-conversion rule looks something like</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>funsplit</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">[</mo><mi>λ</mi><mi>x</mi><mo>.</mo><mi>b</mi><mo stretchy="false">/</mo><mi>g</mi><mo stretchy="false">]</mo><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><msub><mo>→</mo> <mi>η</mi></msub><mspace width="thickmathspace"></mspace><mi>c</mi><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">/</mo><mi>g</mi><mo stretchy="false">]</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> funsplit(c[\lambda x.b / g], f) \;\to_\eta\; c[f/g]. </annotation></semantics></math></div> <p>Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo lspace="verythinmathspace">:</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⊢</mo><mi>c</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">g\colon \prod_{x\colon A} B(x) \vdash c\colon C</annotation></semantics></math> is a term containing a free variable of type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\prod_{x\colon A} B(x)</annotation></semantics></math>. By substituting <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>b</mi></mrow><annotation encoding="application/x-tex">\lambda x.b</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math>, we obtain a term of type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> which depends on “a term <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo lspace="verythinmathspace">:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">b\colon B(x)</annotation></semantics></math> containing a free variable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">x\colon A</annotation></semantics></math>”. We then apply the positive eliminator at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f\colon \prod_{x\colon A} B(x)</annotation></semantics></math>, and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math>-rule says that this can be computed by just substituting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>.</p> <h3 id="positive_versus_negative">Positive versus negative</h3> <p>As usual, the positive and negative formulations are equivalent in a suitable sense. They have the same constructor, while we can formulate the eliminators in terms of each other:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≔</mo><mi>funsplit</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">[</mo><mi>a</mi><mo stretchy="false">/</mo><mi>x</mi><mo stretchy="false">]</mo><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mi>funsplit</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≔</mo><mi>c</mi><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>b</mi><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} f(a) &\coloneqq funsplit(b[a/x], f)\\ funsplit(c, f) &\coloneqq c[f(x)/b] \end{aligned} </annotation></semantics></math></div> <p>The conversion rules also correspond.</p> <p>In dependent type theory, this definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>funsplit</mi></mrow><annotation encoding="application/x-tex">funsplit</annotation></semantics></math> only gives us a properly typed dependent eliminator if the negative dependent product type satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math>-conversion. As usual, if it satisfies <a href="/nlab/show/eta-conversion#Propositional">propositional eta-conversion</a> then we can transport along that instead—and conversely, the dependent eliminator allows us to prove propositional <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math>-conversion. This is the content of Propositions 3.5, 3.6, and 3.7 in <a href="#GarnerSDP">(Garner)</a>.</p> <h3 id="dependent_product_types_a_la_russell_and_a_la_tarski">Dependent product types a la Russell and a la Tarski</h3> <p>In <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a>, there are two different ways to interpret the term <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f:\prod_{x:A} B(x)</annotation></semantics></math>:</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is literally a family of terms in the family of types <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(x)</annotation></semantics></math> indexed by <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></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is a term representation for a family of terms in the family of types <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(x)</annotation></semantics></math> indexed by <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></p> </li> </ol> <p>This situation is similar to how there are two notions of <a class="existingWikiWord" href="/nlab/show/type+universe">type universe</a> where <a class="existingWikiWord" href="/nlab/show/small+types">small types</a> of a universe are interpreted <a class="existingWikiWord" href="/nlab/show/Russell+universe">a la Russell</a>, literally as types, or <a class="existingWikiWord" href="/nlab/show/Tarski+universe">a la Tarski</a>, as a term representation of types. Thus, in analogy to type universes, we can refer to dependent product types <em>a la Russell</em> and function types <em>a la Tarski</em>.</p> <p>Dependent product types a la Russell and a la Tarski are expressed respectively via the <a class="existingWikiWord" href="/nlab/show/elimination+rule">elimination rule</a> of function types:</p> <ul> <li>given 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> and the type family <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>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A \vdash B(x)</annotation></semantics></math> and an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f:\prod_{x:A} B(x)</annotation></semantics></math>, one could form the family of terms <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>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">x:A \vdash f(x):B</annotation></semantics></math></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>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></mrow><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>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma \vdash f:\prod_{x:A} B(x)}{\Gamma, x:A \vdash f(x):B(x)}</annotation></semantics></math></div> <ul> <li>given 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> and the type family <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>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A \vdash B(x)</annotation></semantics></math> one could form the family of terms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi mathvariant="normal">eval</mi><mo stretchy="false">(</mo><mi>f</mi><mo>,</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">f:\prod_{x:A} B(x), x:A \vdash \mathrm{eval}(f, x):B(x)</annotation></semantics></math></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>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></mrow><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><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi mathvariant="normal">eval</mi><mo stretchy="false">(</mo><mi>f</mi><mo>,</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{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma, f:\prod_{x:A} B(x), x:A \vdash \mathrm{eval}(f, x):B(x)}</annotation></semantics></math></div> <p>Dependent product types a la Tarski corresponds to the notion of <a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a> in <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> where the dependent product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi(A, B)</annotation></semantics></math> literally comes with a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">eval</mi><mo>:</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>×</mo><mi>A</mi><mo>→</mo><mi>Σ</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{eval}:\Pi(A, B) \times A \to \Sigma(A, B)</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/slice+category">slice category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">/</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">C/A</annotation></semantics></math>, but dependent product types a la Russell are the one most commonly used in <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a>.</p> <p>The conversion rules for dependent product types a la Russell are as follows:</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>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></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><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><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></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash b(x):B(x)}{\Gamma, x:A \vdash (\lambda x:A.b(x))(x) \equiv b(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><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>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></mrow><mrow><mi>Γ</mi><mo>⊢</mo><mi>f</mi><mo>≡</mo><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><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 A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma \vdash f:\prod_{x:A} B(x)}{\Gamma \vdash f \equiv \lambda x:A.f(x):\prod_{x:A} B(x)}</annotation></semantics></math></div> <p>and the conversion rules for dependent product types a la Tarski are as follows:</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>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></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi mathvariant="normal">eval</mi><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>,</mo><mi>x</mi><mo stretchy="false">)</mo><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></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash b(x):B(x)}{\Gamma, x:A \vdash \mathrm{eval}(\lambda x:A.b(x), x) \equiv b(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><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>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></mrow><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><mo>⊢</mo><mi>f</mi><mo>≡</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi mathvariant="normal">eval</mi><mo stretchy="false">(</mo><mi>f</mi><mo>,</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 A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma, f:\prod_{x:A} B(x) \vdash f \equiv \lambda x:A.\mathrm{eval}(f, x):\prod_{x:A} B(x)}</annotation></semantics></math></div> <p>For the rest of the article we shall assume the use of dependent product types a la Russell.</p> <h3 id="weak_and_strict_dependent_product_types">Weak and strict dependent product types</h3> <p>In <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a>, <strong>weak dependent product types</strong> are dependent product types for which the <a class="existingWikiWord" href="/nlab/show/computation+rules">computation rules</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math>-conversion) and <a class="existingWikiWord" href="/nlab/show/uniqueness+rules">uniqueness rules</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math>-conversion) are propositional rather than judgmental:</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>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></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>a</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><msubsup><mi>β</mi> <mrow><mi>A</mi><mo>→</mo><mi>B</mi></mrow> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>b</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</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><mo stretchy="false">(</mo><mi>a</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>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, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash b(x):B(x)}{\Gamma, a:A \vdash \beta_{A \to B}^{x:A.b(x)}(a):(\lambda(x:A).b(x))(a) =_{B(x)} 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>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></mrow><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><mo>⊢</mo><msub><mi>η</mi> <mrow><mi>A</mi><mo>→</mo><mi>B</mi></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>:</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>λ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo stretchy="false">)</mo><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{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma, f:\prod_{x:A} B(x) \vdash \eta_{A \to B}(f):f =_{\prod_{x:A} B(x)} \lambda(x:A).f(x)}</annotation></semantics></math></div> <p>Weak dependent product types appear in <a class="existingWikiWord" href="/nlab/show/weak+type+theories">weak type theories</a>.</p> <p>This contrasts with <strong>strict dependent product types</strong> which use judgmental computation and uniqueness rules:</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>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></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>a</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mo stretchy="false">(</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><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>x</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash b(x):B(x)}{\Gamma, a:A \vdash (\lambda(x:A).b(x))(a) \equiv b(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><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>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></mrow><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><mo>⊢</mo><mi>f</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>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 A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma, f:\prod_{x:A} B(x) \vdash f \equiv \lambda(x:A).f(x):\prod_{x:A} B(x)}</annotation></semantics></math></div> <p>Strict dependent product types appear in most dependent type theories such as <a class="existingWikiWord" href="/nlab/show/Martin-L%C3%B6f+type+theory">Martin-Löf type theory</a>, <a class="existingWikiWord" href="/nlab/show/observational+type+theory">observational type theory</a>, and <a class="existingWikiWord" href="/nlab/show/cubical+type+theory">cubical type theory</a>.</p> <p>For strict dependent product types, the judgmental computation and uniqueness rules automatically imply the propositional computation and uniqueness rules, as by the rules for <a class="existingWikiWord" href="/nlab/show/judgmental+equality">judgmental equality</a> and <a class="existingWikiWord" href="/nlab/show/identity+types">identity types</a>, judgmental equality of two terms always implies propositional equality of the two terms.</p> <h3 id="in_terms_of_function_types">In terms of function types</h3> <p>Given a <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a> with <a class="existingWikiWord" href="/nlab/show/function+types">function types</a>, <a class="existingWikiWord" href="/nlab/show/dependent+sum+types">dependent sum types</a>, and <a class="existingWikiWord" href="/nlab/show/identity+types">identity types</a>, the dependent product type of a type family <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(x)</annotation></semantics></math> indexed by <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> can be defined as the type of functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f:A \to \sum_{x:A} B(x)</annotation></semantics></math> from <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> to the dependent sum type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\sum_{x:A} B(x)</annotation></semantics></math> such that the composite of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> with the first projection function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo>:</mo><mrow><mo>(</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\pi_1:\left(\sum_{x:A} B(x)\right) \to A</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/identity+function">identity function</a> on <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></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><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><mo>≔</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>f</mi><mo>:</mo><mi>A</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>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>A</mi><mo>→</mo><mi>A</mi></mrow></msub><msub><mi mathvariant="normal">id</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\prod_{x:A} B(x) \coloneqq \sum_{f:A \to \sum_{x:A} B(x)} \lambda x:A.\pi_1(f(x)) =_{A \to A} \mathrm{id}_A</annotation></semantics></math></div> <p>The underlying family of elements is then given by the composite of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f:A \to \sum_{x:A} B(x)</annotation></semantics></math> with the second projection function of the dependent sum type:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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">x:A \vdash \pi_2(f(x)):B(x)</annotation></semantics></math></div> <p>Similarly, given a family of elements <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>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></mrow><annotation encoding="application/x-tex">x:A \vdash b(x):B(x)</annotation></semantics></math>, one could construct the function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>b</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><mi>A</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><annotation encoding="application/x-tex">\lambda x:A.(x, b(x)):A \to \sum_{x:A} B(x)</annotation></semantics></math></div> <p>such that given <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>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><mo stretchy="false">(</mo><mi>x</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><mo stretchy="false">)</mo><mo>≡</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\pi_1(\lambda x:A.(x, b(x))(x)) \equiv x</annotation></semantics></math>. By lambda abstraction, one has</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><mo stretchy="false">(</mo><mi>x</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><mo stretchy="false">)</mo><mo>≡</mo><msub><mi mathvariant="normal">id</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\lambda x:A.\pi_1(\lambda x:A.(x, b(x))(x)) \equiv \mathrm{id}_A</annotation></semantics></math></div> <p>and so the dependent function is given by</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 stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>b</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">refl</mi> <mrow><mi>A</mi><mo>→</mo><mi>A</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi mathvariant="normal">id</mi> <mi>A</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>f</mi><mo>:</mo><mi>A</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>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>A</mi><mo>→</mo><mi>A</mi></mrow></msub><msub><mi mathvariant="normal">id</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">(\lambda x:A.(x, b(x)), \mathrm{refl}_{A \to A}(\mathrm{id}_A)):\sum_{f:A \to \sum_{x:A} B(x)} \lambda x:A.\pi_1(f(x)) =_{A \to A} \mathrm{id}_A</annotation></semantics></math></div> <p>One also has <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><mo stretchy="false">(</mo><mi>x</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><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">\pi_2(\lambda x:A.(x, b(x))(x)) \equiv b(x)</annotation></semantics></math> which is the associated computation rule for dependent function types. Meanwhile, from the judgmental <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math>-conversion rules for negative dependent sum types and function types, one could prove the judgmental <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math>-conversion rule for dependent function types. Given</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>f</mi><mo>:</mo><mi>A</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>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>A</mi><mo>→</mo><mi>A</mi></mrow></msub><msub><mi mathvariant="normal">id</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">f:\sum_{f:A \to \sum_{x:A} B(x)} \lambda x:A.\pi_1(f(x)) =_{A \to A} \mathrm{id}_A</annotation></semantics></math></div> <p>one has</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>≡</mo><mo stretchy="false">(</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f \equiv (\lambda x:A.(x, \pi_2(f(x))), \pi_2(f))</annotation></semantics></math></div> <h3 id="as_types_of_dependent_anafunctions">As types of dependent anafunctions</h3> <p>In the same way that one could define <a class="existingWikiWord" href="/nlab/show/equivalence+types">equivalence types</a> as types of <a class="existingWikiWord" href="/nlab/show/one-to-one+correspondences">one-to-one correspondences</a> and <a class="existingWikiWord" href="/nlab/show/function+types">function types</a> as types of <a class="existingWikiWord" href="/nlab/show/anafunctions">anafunctions</a>, one could define dependent function types as types of <a class="existingWikiWord" href="/nlab/show/dependent+anafunctions">dependent anafunctions</a>. This requires both <a class="existingWikiWord" href="/nlab/show/identity+types">identity types</a> and <a class="existingWikiWord" href="/nlab/show/heterogeneous+identity+types">heterogeneous identity types</a> being defined first, which we shall write as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>b</mi></mrow><annotation encoding="application/x-tex">a =_A b</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><msubsup><mo>=</mo> <mi>B</mi> <mi>p</mi></msubsup><mi>y</mi></mrow><annotation encoding="application/x-tex">x =_{B}^{p} y</annotation></semantics></math> respectively for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>:</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a:A</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>:</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">b:A</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>a</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>b</mi></mrow><annotation encoding="application/x-tex">p:a =_A b</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:B(a)</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y:B(b)</annotation></semantics></math>. We use Agda notation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><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></mrow><annotation encoding="application/x-tex">(x:A) \to B(x)</annotation></semantics></math> for dependent function types rather than the dependent product type notation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\prod_{x:A} B(x)</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</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></mrow><annotation encoding="application/x-tex">\Pi(x:A).B(x)</annotation></semantics></math> in this section.</p> <p>Rules for dependent function types</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>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></mrow><mrow><mi>Γ</mi><mo>⊢</mo><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><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow></mfrac><mspace width="2em"></mspace><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>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></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>f</mi><mo>:</mo><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>,</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><msub><mi>ℱ</mi> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</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, x:A \vdash B(x) \; \mathrm{type}}{\Gamma \vdash (x:A) \to B(x) \; \mathrm{type}} \qquad \frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma, f:(x:A) \to B(x), x:A, y:B(x) \vdash \mathcal{F}_{A, B}(f, x, y) \; \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><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>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>f</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>λ</mi><mi>x</mi><mo>.</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><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></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash f(x):B(x)}{\Gamma \vdash \lambda x.f(x):(x:A) \to 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><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>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>f</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>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>α</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><msub><mi>ℱ</mi> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msub><mo stretchy="false">(</mo><mi>λ</mi><mi>x</mi><mo>.</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><mi>f</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 \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash f(x):B(x)}{\Gamma, x:A \vdash \alpha(x):\mathcal{F}_{A, B}(\lambda x.f(x), x, f(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>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></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>f</mi><mo>:</mo><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>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi mathvariant="normal">ev</mi><mo stretchy="false">(</mo><mi>f</mi><mo>,</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{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma, f:(x:A) \to B(x), x:A \vdash \mathrm{ev}(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><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>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></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>f</mi><mo>:</mo><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>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>β</mi><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><msub><mi>ℱ</mi> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi mathvariant="normal">ev</mi><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma, f:(x:A) \to B(x), x:A \vdash \beta(f, x):\mathcal{F}_{A, B}(f, x, \mathrm{ev}(f, 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>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></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>f</mi><mo>:</mo><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>,</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>u</mi><mo>:</mo><msub><mi>ℱ</mi> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>⊢</mo><mi>κ</mi><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>u</mi><mo stretchy="false">)</mo><mo>:</mo><mi mathvariant="normal">ev</mi><mo stretchy="false">(</mo><mi>f</mi><mo>,</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>y</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma, f:(x:A) \to B(x), x:A, y:B(x), u:\mathcal{F}_{A, B}(f, x, y) \vdash \kappa(f, x, y, u):\mathrm{ev}(f, x) =_{B(x)} 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><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></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>f</mi><mo>:</mo><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>,</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>u</mi><mo>:</mo><msub><mi>ℱ</mi> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>⊢</mo><mi>η</mi><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>u</mi><mo stretchy="false">)</mo><mo>:</mo><mi>β</mi><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><msubsup><mo>=</mo> <mrow><msub><mi>ℱ</mi> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow> <mrow><mi>κ</mi><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>u</mi><mo stretchy="false">)</mo></mrow></msubsup><mi>u</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma, f:(x:A) \to B(x), x:A, y:B(x), u:\mathcal{F}_{A, B}(f, x, y) \vdash \eta(f, x, y, u):\beta(f, x) =_{\mathcal{F}_{A, B}(f, x)}^{\kappa(f, x, y, u)} u}</annotation></semantics></math></div> <p>By the rules for <a class="existingWikiWord" href="/nlab/show/dependent+pair+types">dependent pair types</a> and <a class="existingWikiWord" href="/nlab/show/dependent+function+types">dependent function types</a>, one could derive that</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>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></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>f</mi><mo>:</mo><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>⊢</mo><mi>η</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi mathvariant="normal">isContr</mi><mrow><mo>(</mo><mo stretchy="false">(</mo><mi>y</mi><mo>:</mo><mi>B</mi><mo stretchy="false">)</mo><mo>×</mo><msub><mi>ℱ</mi> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma, f:(x:A) \to B(x) \vdash \eta(f):(x:A) \to \mathrm{isContr}\left((y:B) \times \mathcal{F}_{A, B}(f, x, y)\right)}</annotation></semantics></math></div> <p>which is precisely the statement that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℱ</mi> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{F}_{A, B}(f)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/dependent+anafunction">dependent anafunction</a> for all dependent functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><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></mrow><annotation encoding="application/x-tex">f:(x:A) \to B(x)</annotation></semantics></math>.</p> <h2 id="properties">Properties</h2> <h3 id="universal_property_of_dependent_product_types">Universal property of dependent product types</h3> <p>The <strong>universal property of dependent product types</strong> states that for all types <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 type families <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>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A \vdash B(x)</annotation></semantics></math>, there is a family of functions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>A</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><mrow><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></mrow><mo>→</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_A:\prod_{x:A} \left(\prod_{x:A} B(x)\right) \to B(x)</annotation></semantics></math></div> <p>such that for any other type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> and family of functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>C</mi><mo>→</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f:\prod_{x:A} C \to B(x)</annotation></semantics></math>, there is a unique function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>:</mo><mi>C</mi><mo>→</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c:C \to \prod_{x:A} B(x)</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><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>C</mi></mrow></munder><msub><mi>π</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><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>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\prod_{x:A} \prod_{y:C} \pi_A(x)(c(y)) =_{B(x)} f(x)(y)</annotation></semantics></math></div> <p>If there is a <a class="existingWikiWord" href="/nlab/show/type+universe">type universe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>, then one could wrap this into a single <a class="existingWikiWord" href="/nlab/show/axiom">axiom</a>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">up</mi> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>C</mi><mo>:</mo><mi>U</mi></mrow></munder><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>C</mi><mo>→</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></munder><mo>∃</mo><mo>!</mo><mi>c</mi><mo>:</mo><mi>C</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></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>y</mi><mo>:</mo><mi>C</mi></mrow></munder><msub><mi>π</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><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>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{up}_{A, B}:\prod_{C:U} \prod_{f:\prod_{x:A} C \to B(x)} \exists!c:C \to \prod_{x:A} B(x).\prod_{x:A} \prod_{y:C} \pi_A(x)(c(y)) =_{B(x)} f(x)(y)</annotation></semantics></math></div> <h3 id="typal_computation_and_uniqueness_rules">Typal computation and uniqueness rules</h3> <p>The typal computation rule for function types is provable from the other four typal type formers of function types. Given 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>, type family <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>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A \vdash B(x)</annotation></semantics></math> and dependent function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f:\prod_{x:A} B(x)</annotation></semantics></math>, we have, by the elimination rule and the introduction rule, a dependent function <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>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lambda x:A.f(x):\prod_{x:A} B(x)</annotation></semantics></math>, which by the uniqueness rules of dependent product types are equal to each other</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <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><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>:</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>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><annotation encoding="application/x-tex">\eta_{\prod_{x:A} B(x)}(f):f =_{\prod_{x:A} B(x)} \lambda x:A.f(x)</annotation></semantics></math></div> <p>By the inductively defined function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">idtohomotopy</mi></mrow><annotation encoding="application/x-tex">\mathrm{idtohomotopy}</annotation></semantics></math> which takes <a class="existingWikiWord" href="/nlab/show/identifications">identifications</a> between dependent functions to <a class="existingWikiWord" href="/nlab/show/homotopies">homotopies</a> between dependent functions, we have that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">idtohomotopy</mi><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><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><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>η</mi> <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><mo stretchy="false">(</mo><mi>f</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>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><mo stretchy="false">(</mo><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><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{idtohomotopy}(f, \lambda x:A.f(x))(\eta_{\prod_{x:A} B(x)}(f)):\prod_{x:A} f(x) =_{B(x)} (\lambda x:A.f(x))(x)</annotation></semantics></math></div> <p>which is the typal computation rule for dependent function types.</p> <h3 id="TypalCongruenceRules">Typal congruence rules</h3> <p>These are called <em>typal congruence rules</em> because they are the analogue of the judgmental <a class="existingWikiWord" href="/nlab/show/congruence+rules">congruence rules</a> which use <a class="existingWikiWord" href="/nlab/show/identity+types">identity types</a> and <a class="existingWikiWord" href="/nlab/show/weak+equivalence+types">weak equivalence types</a> instead of <a class="existingWikiWord" href="/nlab/show/judgmental+equality">judgmental equality</a>.</p> <h4 id="strict_dependent_product_types">Strict dependent product types</h4> <p>Since <a class="existingWikiWord" href="/nlab/show/dependent+product+types">dependent product types</a> are negative types, we first present the typal congruence rule for the elimination rule of dependent product types</p> <p> <div class='num_theorem'> <h6>Theorem</h6> <p>Given a 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> and a type family <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>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A \vdash B(x)</annotation></semantics></math>, dependent functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f:\prod_{x:A} B(x)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g:\prod_{x:A} B(x)</annotation></semantics></math> and an identification <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>f</mi><msub><mo>=</mo> <mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mi>g</mi></mrow><annotation encoding="application/x-tex">p:f =_{\prod_{x:A} B(x)} g</annotation></semantics></math> there are families of identifications <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 mathvariant="normal">compelim</mi><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><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><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><annotation encoding="application/x-tex">x:A \vdash \mathrm{compelim}(f, g, p)(x):f(x) =_{B(x)} g(x)</annotation></semantics></math>.</p> </div> </p> <p> <div class='proof'> <h6>Proof</h6> <p>We simply define the dependent function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">compelim</mi></mrow><annotation encoding="application/x-tex">\mathrm{compelim}</annotation></semantics></math> to be happly, which is inductively defined on <a class="existingWikiWord" href="/nlab/show/identity+types">identity types</a>.</p> </div> </p> <p>The next is the typal congruence rule for the introduction rule of dependent function types. However, unlike the case for the other two rules, one needs dependent function extensionality.</p> <p> <div class='num_theorem'> <h6>Theorem</h6> <p>Assuming dependent <a class="existingWikiWord" href="/nlab/show/function+extensionality">function extensionality</a>, given 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> and family of types <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>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A \vdash B(x)</annotation></semantics></math>, families of elements <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>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></mrow><annotation encoding="application/x-tex">x:A \vdash b(x):B(x)</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>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></mrow><annotation encoding="application/x-tex">x:A \vdash b'(x):B(x)</annotation></semantics></math>, and families of <a class="existingWikiWord" href="/nlab/show/identifications">identifications</a> <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>p</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><msub><mo>=</mo> <mrow><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mi>b</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A \vdash p(x):b(x) =_{B(x)} b'(x)</annotation></semantics></math>, there is a identification</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">congintro</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><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><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><mo stretchy="false">(</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo stretchy="false">)</mo><mo>.</mo><mi>b</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{congintro}_{x:A.p(x)}:\lambda (x:A).b(x) =_{\prod_{x:A} B(x)} \lambda (x:A).b'(x)</annotation></semantics></math></div> <p></p> </div> </p> <p> <div class='proof'> <h6>Proof</h6> <p>By the computation rule of strict dependent function types, there are families of judgmental equalities</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mo stretchy="false">(</mo><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><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></mrow><annotation encoding="application/x-tex">x:A \vdash ((\lambda x:A.b(x))(x) \equiv b(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><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mo stretchy="false">(</mo><mo stretchy="false">(</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 stretchy="false">)</mo><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></mrow><annotation encoding="application/x-tex">x:A \vdash ((\lambda x:A.b'(x))(x) \equiv b'(x):B(x)</annotation></semantics></math></div> <p>Thus, by the <a class="existingWikiWord" href="/nlab/show/structural+rules">structural rules</a> of <a class="existingWikiWord" href="/nlab/show/judgmental+equality">judgmental equality</a>, there are families of identifications</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><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><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>′</mo><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><annotation encoding="application/x-tex">x:A \vdash p(x):(\lambda x:A.b(x))(x) =_{B(x)} (\lambda x:A.b'(x))(x)</annotation></semantics></math></div> <p>and by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math>-abstraction, one gets the dependent function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo stretchy="false">)</mo><mo>.</mo><mi>p</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><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><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>′</mo><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><annotation encoding="application/x-tex">\lambda (x:A).p(x):\prod_{x:A} (\lambda x:A.b(x))(x) =_{B(x)} (\lambda x:A.b'(x))(x)</annotation></semantics></math></div> <p>By dependent <a class="existingWikiWord" href="/nlab/show/function+extensionality">function extensionality</a>, there is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+types">equivalence of types</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi mathvariant="normal">ext</mi> <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> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>:</mo><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><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><mo stretchy="false">(</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 stretchy="false">)</mo><mo>≃</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><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><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>′</mo><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><annotation encoding="application/x-tex">\mathrm{ext}_{\prod_{x:A} B(x)}^{-1}:(\lambda x:A.b(x)) =_{\prod_{x:A} B(x)} (\lambda x:A.b'(x)) \simeq \prod_{x:A} (\lambda x:A.b(x))(x) =_{B(x)} (\lambda x:A.b'(x))(x)</annotation></semantics></math></div> <p>which yields an identification</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi mathvariant="normal">ext</mi> <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> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><msup><mo></mo><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo stretchy="false">)</mo><mo>.</mo><mi>p</mi><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>b</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><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><mo stretchy="false">(</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 stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{ext}_{\prod_{x:A} B(x)}^{-1}^{-1}(\lambda (x:A).p(x)):(\lambda x:A.b(x)) =_{\prod_{x:A} B(x)} (\lambda x:A.b'(x))</annotation></semantics></math></div> <p>We define</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">congintro</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo>≔</mo><msubsup><mi mathvariant="normal">ext</mi> <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> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><msup><mo></mo><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo stretchy="false">)</mo><mo>.</mo><mi>p</mi><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>b</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><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><mo stretchy="false">(</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 stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{congintro}_{x:A.p(x)} \coloneqq \mathrm{ext}_{\prod_{x:A} B(x)}^{-1}^{-1}(\lambda (x:A).p(x)):(\lambda x:A.b(x)) =_{\prod_{x:A} B(x)} (\lambda x:A.b'(x))</annotation></semantics></math></div> <p></p> </div> </p> <p>Finally, we present the typal congruence rule for the formation rule of function types, which relies upon the previous two results. The theorem and proof differs significantly whether one uses <a class="existingWikiWord" href="/nlab/show/definitional+isomorphisms">definitional isomorphisms</a>, some notion of <a class="existingWikiWord" href="/nlab/show/equivalences+of+types">equivalences of types</a>, or in a <a class="existingWikiWord" href="/nlab/show/dependent+type+theory+with+type+variables">dependent type theory with type variables</a>, <a class="existingWikiWord" href="/nlab/show/identity+type#IdentityTypesBetweenTypes">identity types between types</a></p> <h5 id="using_definitional_isomorphisms">Using definitional isomorphisms</h5> <p> <div class='num_theorem'> <h6>Theorem</h6> <p>Given types <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> and type families <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>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A \vdash B(x)</annotation></semantics></math>, <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>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A' \vdash B'(x)</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/definitional+isomorphisms">definitional isomorphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>A</mi></msub><mo>:</mo><mi>A</mi><mo>≅</mo><mi>A</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">e_A:A \cong A'</annotation></semantics></math> and dependent function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>B</mi></msub><mo>:</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><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><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e_B:\prod_{x:A} B(x) \cong B'(e_A(x))</annotation></semantics></math> consisting of a family of definitional isomorphisms, there is a definitional isomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo>:</mo><mrow><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></mrow><mo>≅</mo><mrow><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><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\mathrm{congform}(e_A, e_B):\left(\prod_{x:A} B(x)\right) \cong \left(\prod_{x:A'} B'(x)\right)</annotation></semantics></math></div> <p></p> </div> </p> <p> <div class='proof'> <h6>Proof</h6> <p>Since for <a class="existingWikiWord" href="/nlab/show/definitional+isomorphism">definitional isomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>A</mi></msub><mo>:</mo><mi>A</mi><mo>≅</mo><mi>A</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">e_A:A \cong A'</annotation></semantics></math>, we have judgmental equalities <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><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><mo>′</mo></mrow><annotation encoding="application/x-tex">e_A(e_A^{-1}(x)) \equiv x:A'</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></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><annotation encoding="application/x-tex">e_A^{-1}(e_A(x)) \equiv x:A</annotation></semantics></math>, so we do not need to transport across <a class="existingWikiWord" href="/nlab/show/identifications">identifications</a>. Instead, we define the function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo>:</mo><mrow><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></mrow><mo>→</mo><mrow><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><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\mathrm{congform}(e_A, e_B):\left(\prod_{x:A} B(x)\right) \to \left(\prod_{x:A'} B'(x)\right)</annotation></semantics></math></div> <p>by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo>≔</mo><mi>λ</mi><mo stretchy="false">(</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><mo stretchy="false">)</mo><mo>.</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{congform}(e_A, e_B) \coloneqq \lambda (f:\prod_{x:A} B(x)).\lambda x:A'.e_B(e_A^{-1}(x))(f(e_A^{-1}(x)))</annotation></semantics></math></div> <p>and the inverse function by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>≔</mo><mi>λ</mi><mo stretchy="false">(</mo><mi>f</mi><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><mo stretchy="false">)</mo><mo>.</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{congform}(e_A, e_B)^{-1} \coloneqq \lambda (f:\prod_{x:A'} B'(x)).\lambda x:A.e_B(x)^{-1}(f(e_A(x)))</annotation></semantics></math></div> <p>Now it suffices to construct judgmental equalities</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><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><mo>⊢</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><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><annotation encoding="application/x-tex">f:\prod_{x:A} B(x) \vdash \mathrm{congform}(e_A, e_B)^{-1}(\mathrm{congform}(e_A, e_B)(f)) \equiv f:\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><mi>g</mi><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><mo>⊢</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><mi>g</mi><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></mrow><annotation encoding="application/x-tex">g:\prod_{x:A'} B'(x) \vdash \mathrm{congform}(e_A, e_B)(\mathrm{congform}(e_A, e_B)^{-1}(g)) \equiv g:\prod_{x:A'} B'(x)</annotation></semantics></math></div> <p>from where it implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{congform}(e_A, e_B)</annotation></semantics></math> is thus a <a class="existingWikiWord" href="/nlab/show/definitional+isomorphism">definitional isomorphism</a>.</p> <p>By definition, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{congform}(e_A, e_B)^{-1}(\mathrm{congform}(e_A, e_B)(f)) \equiv \lambda x:A.e_B(x)^{-1}((\lambda x:A'.e_B(e_A^{-1}(x))(f(e_A^{-1}(x))) )(e_A(x)))</annotation></semantics></math></div> <p>and by the computation rules of strict dependent product types, we have</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><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\lambda x:A'.e_B(e_A^{-1}(x))(f(e_A^{-1}(x))) )(e_A(x)) \equiv e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x))))</annotation></semantics></math></div> <p>and because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">e_A</annotation></semantics></math> is a definitional isomorphism, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x)))) \equiv e_B(x)(f(x))</annotation></semantics></math></div> <p>By the congruence rules for substitution of judgmental equality, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lambda x:A.e_B(x)^{-1}((\lambda x:A'.e_B(e_A^{-1}(x))(f(e_A^{-1}(x))) )(e_A(x))) \equiv \lambda x:A.e_B(x)^{-1}(e_B(x)(f(x)))</annotation></semantics></math></div> <p>Since for all <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> each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e_B(x)</annotation></semantics></math> is also a strict equality, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e_B(x)^{-1}(e_B(x)(f(x))) \equiv f(x)</annotation></semantics></math></div> <p>by the congruence rules for substitution of judgmental equality, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><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><annotation encoding="application/x-tex">\lambda x:A.e_B(x)^{-1}(e_B(x)(f(x))) \equiv \lambda x:A.f(x)</annotation></semantics></math></div> <p>and by the uniqueness rule of dependent function types we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><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><mo>≡</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">\lambda x:A.f(x) \equiv f</annotation></semantics></math></div> <p>thus by the transitive rule for judgmental equality we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">\mathrm{congform}(e_A, e_B)^{-1}(\mathrm{congform}(e_A, e_B)(f)) \equiv f</annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f:\prod_{x:A} B(x)</annotation></semantics></math></p> <p>Similarly, by definition, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{congform}(e_A, e_B)(\mathrm{congform}(e_A, e_B)^{-1}(f)) \equiv \lambda x:A'.e_B(e_A^{-1}(x))(( \lambda x:A.e_B(x)^{-1}(f(e_A(x))) )(e_A^{-1}(x)))</annotation></semantics></math></div> <p>and by the computation rules for strict dependent product types, we have</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><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">( \lambda x:A.e_B(x)^{-1}(f(e_A(x))) )(e_A^{-1}(x)) \equiv e_B(e_A^{-1}(x))^{-1}(f(e_A(e_A^{-1}(x))))</annotation></semantics></math></div> <p>and because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">e_A</annotation></semantics></math> is a definitional isomorphism, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e_B(e_A^{-1}(x))^{-1}(f(e_A(e_A^{-1}(x)))) \equiv e_B(e_A^{-1}(x))^{-1}(f(x))</annotation></semantics></math></div> <p>By the congruence rules for substitution of judgmental equality, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lambda x:A'.e_B(e_A^{-1}(x))(( \lambda x:A.e_B(x)^{-1}(f(e_A(x))) )(e_A^{-1}(x))) \equiv \lambda x:A'.e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(x)))</annotation></semantics></math></div> <p>Since for all <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> each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e_B(x)</annotation></semantics></math> is also a strict equality, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(x))) \equiv f(x)</annotation></semantics></math></div> <p>by the congruence rules for substitution of judgmental equality, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lambda x:A'.e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(x))) \equiv \lambda x:A'.f(x)</annotation></semantics></math></div> <p>and by the uniqueness rule of dependent function types we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><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><mo>≡</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">\lambda x:A.f(x) \equiv f</annotation></semantics></math></div> <p>thus by the transitive rule for judgmental equality we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">\mathrm{congform}(e_A, e_B)(\mathrm{congform}(e_A, e_B)^{-1}(f)) \equiv f</annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo></mrow></msub><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f:\prod_{x:A'} B'(x)</annotation></semantics></math>.</p> <p>Since we have functions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo>:</mo><mrow><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></mrow><mo>→</mo><mrow><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><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\mathrm{congform}(e_A, e_B):\left(\prod_{x:A} B(x)\right) \to \left(\prod_{x:A'} B'(x)\right)</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>:</mo><mrow><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><mo>)</mo></mrow><mo>→</mo><mrow><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></mrow></mrow><annotation encoding="application/x-tex">\mathrm{congform}(e_A, e_B)^{-1}:\left(\prod_{x:A'} B'(x)\right) \to \left(\prod_{x:A} B(x)\right)</annotation></semantics></math></div> <p>and families of judgmental equalities</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><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><mo>⊢</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><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><annotation encoding="application/x-tex">f:\prod_{x:A} B(x) \vdash \mathrm{congform}(e_A, e_B)^{-1}(\mathrm{congform}(e_A, e_B)(f)) \equiv f:\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><mi>g</mi><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><mo>⊢</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><mi>g</mi><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></mrow><annotation encoding="application/x-tex">g:\prod_{x:A'} B'(x) \vdash \mathrm{congform}(e_A, e_B)(\mathrm{congform}(e_A, e_B)^{-1}(g)) \equiv g:\prod_{x:A'} B'(x)</annotation></semantics></math></div> <p>we could form the definitional isomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">toEquiv</mi><mo stretchy="false">(</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>:</mo><mrow><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><mo>)</mo></mrow><mo>≅</mo><mrow><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></mrow></mrow><annotation encoding="application/x-tex">\mathrm{toEquiv}(\mathrm{congform}(e_A, e_B), \mathrm{congform}(e_A, e_B)^{-1}):\left(\prod_{x:A'} B'(x)\right) \cong \left(\prod_{x:A} B(x)\right)</annotation></semantics></math></div> <p>By a common abuse of notation we denote the definitional isomorphism by the same name as the underlying function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{congform}(e_A, e_B)</annotation></semantics></math>; thus we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo>:</mo><mrow><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></mrow><mo>≅</mo><mrow><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><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\mathrm{congform}(e_A, e_B):\left(\prod_{x:A} B(x)\right) \cong \left(\prod_{x:A'} B'(x)\right)</annotation></semantics></math></div> <p></p> </div> </p> <h5 id="using_weak_equivalences_of_types">Using weak equivalences of types</h5> <p> <div class='num_theorem'> <h6>Theorem</h6> <p>Given types <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> and type families <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>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A \vdash B(x)</annotation></semantics></math>, <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>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A' \vdash B'(x)</annotation></semantics></math> and equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>A</mi></msub><mo>:</mo><mi>A</mi><mo>≃</mo><mi>A</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">e_A:A \simeq A'</annotation></semantics></math> and dependent function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>B</mi></msub><mo>:</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><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><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e_B:\prod_{x:A} B(x) \simeq B'(e_A(x))</annotation></semantics></math>, there is an equivalence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo>:</mo><mrow><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></mrow><mo>≃</mo><mrow><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><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\mathrm{congform}(e_A, e_B):\left(\prod_{x:A} B(x)\right) \simeq \left(\prod_{x:A'} B'(x)\right)</annotation></semantics></math></div> <p></p> </div> </p> <p> <div class='proof'> <h6>Proof</h6> <p>We define the function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo>:</mo><mrow><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></mrow><mo>→</mo><mrow><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><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\mathrm{congform}(e_A, e_B):\left(\prod_{x:A} B(x)\right) \to \left(\prod_{x:A'} B'(x)\right)</annotation></semantics></math></div> <p>by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo>≔</mo><mi>λ</mi><mo stretchy="false">(</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><mo stretchy="false">)</mo><mo>.</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{congform}(e_A, e_B) \coloneqq \lambda (f:\prod_{x:A} B(x)).\lambda x:A'.\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), e_B(e_A^{-1}(x))(f(e_A^{-1}(x))))</annotation></semantics></math></div> <p>and the inverse function by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>≔</mo><mi>λ</mi><mo stretchy="false">(</mo><mi>f</mi><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><mo stretchy="false">)</mo><mo>.</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{congform}(e_A, e_B)^{-1} \coloneqq \lambda (f:\prod_{x:A'} B'(x)).\lambda x:A.e_B(x)^{-1}(f(e_A(x)))</annotation></semantics></math></div> <p>where the equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>A</mi></msub><mo>:</mo><mi>A</mi><mo>≃</mo><mi>A</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">e_A:A \simeq A'</annotation></semantics></math> has families of identifications</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>′</mo><mo>:</mo><mi>A</mi><mo>⊢</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>A</mi><mo>′</mo></mrow></msub><mi>x</mi></mrow><annotation encoding="application/x-tex">x':A \vdash \mathrm{sec}_{e_A}(x):e_A(e_A^{-1}(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><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi></mrow><annotation encoding="application/x-tex">x:A \vdash \mathrm{ret}_{e_A}(x):e_A^{-1}(e_A(x)) =_A x</annotation></semantics></math></div> <p>witnessing that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">e_A^{-1}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/section">section</a> and <a class="existingWikiWord" href="/nlab/show/retraction">retraction</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">e_A</annotation></semantics></math> respectively.</p> <p>Now it suffices to construct families of identifications</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><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><mo>⊢</mo><msub><mi mathvariant="normal">ret</mi> <mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>:</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><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>f</mi></mrow><annotation encoding="application/x-tex">f:\prod_{x:A} B(x) \vdash \mathrm{ret}_{\mathrm{congform}(e_A, e_B)}(f):\mathrm{congform}(e_A, e_B)^{-1}(\mathrm{congform}(e_A, e_B)(f)) =_{\prod_{x:A} B(x)} f</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>g</mi><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><mo>⊢</mo><msub><mi mathvariant="normal">sec</mi> <mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>:</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><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></mrow></msub><mi>g</mi></mrow><annotation encoding="application/x-tex">g:\prod_{x:A'} B'(x) \vdash \mathrm{sec}_{\mathrm{congform}(e_A, e_B)}(g):\mathrm{congform}(e_A, e_B)(\mathrm{congform}(e_A, e_B)^{-1}(g)) =_{\prod_{x:A'} B'(x)} g</annotation></semantics></math></div> <p>from where it implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{congform}(e_A, e_B)</annotation></semantics></math> has a coherent inverse and contractible fibers and is thus an <a class="existingWikiWord" href="/nlab/show/equivalence+of+types">equivalence of types</a>.</p> <hr /> <p>By definition,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mrow><mi>B</mi><mo>′</mo></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{congform}(e_A, e_B)^{-1}(\mathrm{congform}(e_A, e_B)(f)) \equiv \lambda x:A.e_B(x)^{-1}((\lambda x:A'.\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), e_{B'}(e_A^{-1}(x))(f(e_A^{-1}(x)))))(e_A(x)))</annotation></semantics></math></div> <p>By the computation rules of strict dependent function types, there is a family of judgmental equalities</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mo stretchy="false">(</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A \vdash (\lambda x:A'.\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), e_B(e_A^{-1}(x))(f(e_A^{-1}(x)))))(e_A(x)) \equiv \mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{sec}_{e_A}(e_A(x)), e_B(e_A^{-1}(x))(f(e_A^{-1}(e_A(x)))))</annotation></semantics></math></div> <p>and thus by the structural rules of <a class="existingWikiWord" href="/nlab/show/judgmental+equalities">judgmental equalities</a> and the judgmental congruence rules for dependent function types, a judgmental equality</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{congform}(e_A, e_B)^{-1}(\mathrm{congform}(e_A, e_B)(f)) \equiv \lambda x:A.e_B(x)^{-1}(\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{sec}_{e_A}(e_A(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x))))))</annotation></semantics></math></div> <p>In addition, the equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>A</mi></msub><mo>:</mo><mi>A</mi><mo>≃</mo><mi>A</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">e_A:A \simeq A'</annotation></semantics></math> has the coherence condition</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">coh</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>A</mi><mo>′</mo></mrow></msub><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><msub><mi mathvariant="normal">ap</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{coh}_{e_A}(x):\mathrm{sec}_{e_A}(e_A(x)) =_{e_A(e_A^{-1}(e_A(x))) =_{A'} e_A(x)} \mathrm{ap}_{e_A}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x))</annotation></semantics></math></div> <p>So we have the family of elements</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A \vdash \mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{sec}_{e_A}(e_A(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x))))):B'(e_A(x))</annotation></semantics></math></div> <p>and by applying the function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mi>p</mi><mo>:</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>A</mi><mo>′</mo></mrow></msub><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>.</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>p</mi><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lambda p:e_A(e_A^{-1}(e_A(x))) =_{A'} e_A(x).\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), p, e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x)))))</annotation></semantics></math></div> <p>across <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">coh</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{coh}_{e_A}(x)</annotation></semantics></math> one gets the family of identifications</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><msub><mi mathvariant="normal">ap</mi> <mrow><mi>λ</mi><mi>p</mi><mo>:</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>A</mi><mo>′</mo></mrow></msub><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>.</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>p</mi><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ap</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">coh</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A \vdash \mathrm{ap}_{\lambda p:e_A(e_A^{-1}(e_A(x))) =_{A'} e_A(x).\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), p, e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x)))))}(\mathrm{sec}_{e_A}(e_A(x)), \mathrm{ap}_{e_A}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x)), \mathrm{coh}_{e_A}(x)</annotation></semantics></math></div> <p>in type</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ap</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{sec}_{e_A}(e_A(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x))))) =_{B'(e_A(x)} \mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{ap}_{e_A}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x)))))</annotation></semantics></math></div> <p>From the family of elements</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ap</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A \vdash \mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{ap}_{e_A}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x))))):B(e_A(x))</annotation></semantics></math></div> <p>one could define the family of dependent functions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>λ</mi><mi>y</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>λ</mi><mi>p</mi><mo>:</mo><mi>y</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi><mo>.</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ap</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>y</mi><mo>:</mo><mi>A</mi></mrow></munder><mi>y</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi><mo>)</mo></mrow><mo>→</mo><mi>B</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A \vdash \lambda y:A.\lambda p:y =_A x.\mathrm{transport}_{x:A'.B'(x)}(e_A(y), e_A(x), \mathrm{ap}_{e_A}(y, x, p), e_B(y)(f(y))):\left(\sum_{y:A} y =_A x\right) \to B(e_A(x))</annotation></semantics></math></div> <p>This means by applying the above dependent function to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e_A^{-1}(e_A(x))</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{ret}_{e_A}(x)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">refl</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{refl}_A(x)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{ret}_{e_A}(x)</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">apd</mi> <mrow><mi>y</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>y</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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><annotation encoding="application/x-tex">\mathrm{apd}_{y:A.y =_A x}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x), \mathrm{ret}_{e_A}(x), \mathrm{refl}_A(x))</annotation></semantics></math>, one gets the family of identifications</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><msub><mi mathvariant="normal">apbinary</mi> <mrow><mi>λ</mi><mi>y</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>λ</mi><mi>p</mi><mo>:</mo><mi>y</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi><mo>.</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ap</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">apd</mi> <mrow><mi>y</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>y</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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><annotation encoding="application/x-tex">x:A \vdash \mathrm{apbinary}_{\lambda y:A.\lambda p:y =_A x.\mathrm{transport}_{x:A'.B'(x)}(e_A(y), e_A(x), \mathrm{ap}_{e_A}(y, x, p), e_B(y)(f(y)))}(e_A^{-1}(e_A(x)), \mathrm{ret}_{e_A}(x), x, \mathrm{refl}_A(x), \mathrm{ret}_{e_A}(x), \mathrm{apd}_{y:A.y =_A x}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x), \mathrm{ret}_{e_A}(x), \mathrm{refl}_A(x))</annotation></semantics></math></div> <p>in type</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ap</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>B</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ap</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><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><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{ap}_{e_A}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x))))) =_{B(e_A(x))} \mathrm{transport}_{x:A'.B'(x)}(e_A(x), e_A(x), \mathrm{ap}_{e_A}(x, x, \mathrm{refl}_A(x)), e_B(x)(f(x)))</annotation></semantics></math></div> <p>By the judgmental computation rules of identity types, we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">ap</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><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><mo>≡</mo><msub><mi mathvariant="normal">refl</mi> <mrow><mi>A</mi><mo>′</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{ap}_{e_A}(x, x, \mathrm{refl}_A(x)) \equiv \mathrm{refl}_{A'}(e_A(x))</annotation></semantics></math>, which means that we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ap</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><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><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">refl</mi> <mrow><mi>A</mi><mo>′</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{transport}_{x:A'.B'(x)}(e_A(x), e_A(x), \mathrm{ap}_{e_A}(x, x, \mathrm{refl}_A(x)), e_B(x)(f(x))) \equiv \mathrm{transport}_{x:A'.B'(x)}(e_A(x), e_A(x), \mathrm{refl}_{A'}(e_A(x)), e_B(x)(f(x)))</annotation></semantics></math></div> <p>and we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">refl</mi> <mrow><mi>A</mi><mo>′</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{transport}_{x:A'.B'(x)}(e_A(x), e_A(x), \mathrm{refl}_{A'}(e_A(x)), e_B(x)(f(x))) \equiv e_B(x)(f(x))</annotation></semantics></math></div> <p>Thus by concatenation we have identification</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center"><mtr><mtd><msub><mi mathvariant="normal">ap</mi> <mrow><mi>λ</mi><mi>p</mi><mo>:</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>A</mi><mo>′</mo></mrow></msub><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>.</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>p</mi><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ap</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">coh</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>•</mo><msub><mi mathvariant="normal">apbinary</mi> <mrow><mi>λ</mi><mi>y</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>λ</mi><mi>p</mi><mo>:</mo><mi>y</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi><mo>.</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ap</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">apd</mi> <mrow><mi>y</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>y</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex"> \begin{array}{c} \mathrm{ap}_{\lambda p:e_A(e_A^{-1}(e_A(x))) =_{A'} e_A(x).\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), p, e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x)))))}(\mathrm{sec}_{e_A}(e_A(x)), \mathrm{ap}_{e_A}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x)), \mathrm{coh}_{e_A}(x) \\ \bullet \mathrm{apbinary}_{\lambda y:A.\lambda p:y =_A x.\mathrm{transport}_{x:A'.B'(x)}(e_A(y), e_A(x), \mathrm{ap}_{e_A}(y, x, p), e_B(y)(f(y)))}(e_A^{-1}(e_A(x)), \mathrm{ret}_{e_A}(x), x, \mathrm{refl}_A(x), \mathrm{ret}_{e_A}(x), \mathrm{apd}_{y:A.y =_A x}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x), \mathrm{ret}_{e_A}(x), \mathrm{refl}_A(x)) \end{array} </annotation></semantics></math></div> <p>in type</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{sec}_{e_A}(e_A(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x))))) =_{B'(e_A(x))} e_B(x)(f(x))</annotation></semantics></math></div> <p>and by application of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">e_B(x)^{-1}</annotation></semantics></math> to the above identification, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center"><mtr><mtd><msub><mi mathvariant="normal">ap</mi> <mrow><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow></msub><mo stretchy="false">(</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo></mtd></mtr> <mtr><mtd><msub><mi mathvariant="normal">ap</mi> <mrow><mi>λ</mi><mi>p</mi><mo>:</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>A</mi><mo>′</mo></mrow></msub><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>.</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>p</mi><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ap</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">coh</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>•</mo><msub><mi mathvariant="normal">apbinary</mi> <mrow><mi>λ</mi><mi>y</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>λ</mi><mi>p</mi><mo>:</mo><mi>y</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi><mo>.</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ap</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">apd</mi> <mrow><mi>y</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>y</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex"> \begin{array}{c} \mathrm{ap}_{e_B(x)^{-1}}(\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{sec}_{e_A}(e_A(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x))))), e_B(x)(f(x)), \\ \mathrm{ap}_{\lambda p:e_A(e_A^{-1}(e_A(x))) =_{A'} e_A(x).\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), p, e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x)))))}(\mathrm{sec}_{e_A}(e_A(x)), \mathrm{ap}_{e_A}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x)), \mathrm{coh}_{e_A}(x) \\ \bullet \mathrm{apbinary}_{\lambda y:A.\lambda p:y =_A x.\mathrm{transport}_{x:A'.B'(x)}(e_A(y), e_A(x), \mathrm{ap}_{e_A}(y, x, p), e_B(y)(f(y)))}(e_A^{-1}(e_A(x)), \mathrm{ret}_{e_A}(x), x, \mathrm{refl}_A(x), \mathrm{ret}_{e_A}(x), \mathrm{apd}_{y:A.y =_A x}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x), \mathrm{ret}_{e_A}(x), \mathrm{refl}_A(x)) \end{array} </annotation></semantics></math></div> <p>in type</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e_B(x)^{-1}(\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{sec}_{e_A}(e_A(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x)))))) =_{B(x)} e_B(x)^{-1}(e_B(x)(f(x)))</annotation></semantics></math></div> <p>Similarly, The family of equivalences</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><msub><mi>e</mi> <mi>B</mi></msub><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><mo>≃</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A \vdash e_B(x):B(x) \simeq B'(e_A(x))</annotation></semantics></math></div> <p>has a family of identifications</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>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><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>B</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>:</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><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>y</mi></mrow><annotation encoding="application/x-tex">x:A, y:B(x) \vdash \mathrm{ret}_{e_B}(x, y):e_B(x)^{-1}(e_B(x)(y)) =_{B(x)} y</annotation></semantics></math></div> <p>witnessing that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">e_B(x)^{-1}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/retraction">retraction</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e_B(x)</annotation></semantics></math> for each <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>.</p> <p>By substituting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math> in for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> we get the family of identifications</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>B</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><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>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A \vdash \mathrm{ret}_{e_B}(x, f(x)):e_B(x)^{-1}(e_B(x)(f(x))) =_{B(x)} f(x)</annotation></semantics></math></div> <p>and thus the family of identifications</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center"><mtr><mtd><msub><mi mathvariant="normal">ap</mi> <mrow><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow></msub><mo stretchy="false">(</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo></mtd></mtr> <mtr><mtd><msub><mi mathvariant="normal">ap</mi> <mrow><mi>λ</mi><mi>p</mi><mo>:</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>A</mi><mo>′</mo></mrow></msub><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>.</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>p</mi><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ap</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">coh</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>•</mo><msub><mi mathvariant="normal">apbinary</mi> <mrow><mi>λ</mi><mi>y</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>λ</mi><mi>p</mi><mo>:</mo><mi>y</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi><mo>.</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ap</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">apd</mi> <mrow><mi>y</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>y</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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></mtd></mtr> <mtr><mtd><mo>•</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>B</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex"> \begin{array}{c} \mathrm{ap}_{e_B(x)^{-1}}(\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{sec}_{e_A}(e_A(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x))))), e_B(x)(f(x)), \\ \mathrm{ap}_{\lambda p:e_A(e_A^{-1}(e_A(x))) =_{A'} e_A(x).\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), p, e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x)))))}(\mathrm{sec}_{e_A}(e_A(x)), \mathrm{ap}_{e_A}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x)), \mathrm{coh}_{e_A}(x) \\ \bullet \mathrm{apbinary}_{\lambda y:A.\lambda p:y =_A x.\mathrm{transport}_{x:A'.B'(x)}(e_A(y), e_A(x), \mathrm{ap}_{e_A}(y, x, p), e_B(y)(f(y)))}(e_A^{-1}(e_A(x)), \mathrm{ret}_{e_A}(x), x, \mathrm{refl}_A(x), \mathrm{ret}_{e_A}(x), \mathrm{apd}_{y:A.y =_A x}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x), \mathrm{ret}_{e_A}(x), \mathrm{refl}_A(x)) \\ \bullet \mathrm{ret}_{e_B}(x, f(x)) \end{array} </annotation></semantics></math></div> <p>in type</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><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>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e_B(x)^{-1}(\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{sec}_{e_A}(e_A(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x)))))) =_{B(x)} f(x)</annotation></semantics></math></div> <p>and the dependent function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center"><mtr><mtd><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><msub><mi mathvariant="normal">ap</mi> <mrow><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow></msub><mo stretchy="false">(</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo></mtd></mtr> <mtr><mtd><msub><mi mathvariant="normal">ap</mi> <mrow><mi>λ</mi><mi>p</mi><mo>:</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>A</mi><mo>′</mo></mrow></msub><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>.</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>p</mi><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ap</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">coh</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>•</mo><msub><mi mathvariant="normal">apbinary</mi> <mrow><mi>λ</mi><mi>y</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>λ</mi><mi>p</mi><mo>:</mo><mi>y</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi><mo>.</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ap</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">apd</mi> <mrow><mi>y</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>y</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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></mtd></mtr> <mtr><mtd><mo>•</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>B</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex"> \begin{array}{c} \lambda x:A.\mathrm{ap}_{e_B(x)^{-1}}(\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{sec}_{e_A}(e_A(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x))))), e_B(x)(f(x)), \\ \mathrm{ap}_{\lambda p:e_A(e_A^{-1}(e_A(x))) =_{A'} e_A(x).\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), p, e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x)))))}(\mathrm{sec}_{e_A}(e_A(x)), \mathrm{ap}_{e_A}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x)), \mathrm{coh}_{e_A}(x) \\ \bullet \mathrm{apbinary}_{\lambda y:A.\lambda p:y =_A x.\mathrm{transport}_{x:A'.B'(x)}(e_A(y), e_A(x), \mathrm{ap}_{e_A}(y, x, p), e_B(y)(f(y)))}(e_A^{-1}(e_A(x)), \mathrm{ret}_{e_A}(x), x, \mathrm{refl}_A(x), \mathrm{ret}_{e_A}(x), \mathrm{apd}_{y:A.y =_A x}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x), \mathrm{ret}_{e_A}(x), \mathrm{refl}_A(x)) \\ \bullet \mathrm{ret}_{e_B}(x, f(x)) \end{array} </annotation></semantics></math></div> <p>in type</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><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>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\prod_{x:A} e_B(x)^{-1}(\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{sec}_{e_A}(e_A(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x)))))) =_{B(x)} f(x)</annotation></semantics></math></div> <p>By dependent function extensionality, we get the identification</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center"><mtr><mtd><msubsup><mi mathvariant="normal">ext</mi> <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> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><msub><mi mathvariant="normal">ap</mi> <mrow><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow></msub><mo stretchy="false">(</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo></mtd></mtr> <mtr><mtd><msub><mi mathvariant="normal">ap</mi> <mrow><mi>λ</mi><mi>p</mi><mo>:</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>A</mi><mo>′</mo></mrow></msub><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>.</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>p</mi><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ap</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">coh</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>•</mo><msub><mi mathvariant="normal">apbinary</mi> <mrow><mi>λ</mi><mi>y</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>λ</mi><mi>p</mi><mo>:</mo><mi>y</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi><mo>.</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ap</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">apd</mi> <mrow><mi>y</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>y</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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></mtd></mtr> <mtr><mtd><mo>•</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>B</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex"> \begin{array}{c} \mathrm{ext}_{\prod_{x:A} B(x)}^{-1}(\lambda x:A.\mathrm{ap}_{e_B(x)^{-1}}(\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{sec}_{e_A}(e_A(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x))))), e_B(x)(f(x)), \\ \mathrm{ap}_{\lambda p:e_A(e_A^{-1}(e_A(x))) =_{A'} e_A(x).\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), p, e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x)))))}(\mathrm{sec}_{e_A}(e_A(x)), \mathrm{ap}_{e_A}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x)), \mathrm{coh}_{e_A}(x) \\ \bullet \mathrm{apbinary}_{\lambda y:A.\lambda p:y =_A x.\mathrm{transport}_{x:A'.B'(x)}(e_A(y), e_A(x), \mathrm{ap}_{e_A}(y, x, p), e_B(y)(f(y)))}(e_A^{-1}(e_A(x)), \mathrm{ret}_{e_A}(x), x, \mathrm{refl}_A(x), \mathrm{ret}_{e_A}(x), \mathrm{apd}_{y:A.y =_A x}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x), \mathrm{ret}_{e_A}(x), \mathrm{refl}_A(x)) \\ \bullet \mathrm{ret}_{e_B}(x, f(x))) \end{array} </annotation></semantics></math></div> <p>in type</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><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>f</mi></mrow><annotation encoding="application/x-tex">\lambda x:A.e_B(x)^{-1}(\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{sec}_{e_A}(e_A(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x)))))) =_{\prod_{x:A} B(x)} f</annotation></semantics></math></div> <p>and since we defined</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{congform}(e_A, e_B)^{-1}(\mathrm{congform}(e_A, e_B)(f)) \equiv \lambda x:A.e_B(x)^{-1}(\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{sec}_{e_A}(e_A(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x))))))</annotation></semantics></math></div> <p>we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center"><mtr><mtd><msubsup><mi mathvariant="normal">ext</mi> <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> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><msub><mi mathvariant="normal">ap</mi> <mrow><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow></msub><mo stretchy="false">(</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo></mtd></mtr> <mtr><mtd><msub><mi mathvariant="normal">ap</mi> <mrow><mi>λ</mi><mi>p</mi><mo>:</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>A</mi><mo>′</mo></mrow></msub><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>.</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>p</mi><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ap</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">coh</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>•</mo><msub><mi mathvariant="normal">apbinary</mi> <mrow><mi>λ</mi><mi>y</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>λ</mi><mi>p</mi><mo>:</mo><mi>y</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi><mo>.</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ap</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">apd</mi> <mrow><mi>y</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>y</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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></mtd></mtr> <mtr><mtd><mo>•</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>B</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><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>f</mi></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex"> \begin{array}{c} \mathrm{ext}_{\prod_{x:A} B(x)}^{-1}(\lambda x:A.\mathrm{ap}_{e_B(x)^{-1}}(\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{sec}_{e_A}(e_A(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x))))), e_B(x)(f(x)), \\ \mathrm{ap}_{\lambda p:e_A(e_A^{-1}(e_A(x))) =_{A'} e_A(x).\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), p, e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x)))))}(\mathrm{sec}_{e_A}(e_A(x)), \mathrm{ap}_{e_A}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x)), \mathrm{coh}_{e_A}(x) \\ \bullet \mathrm{apbinary}_{\lambda y:A.\lambda p:y =_A x.\mathrm{transport}_{x:A'.B'(x)}(e_A(y), e_A(x), \mathrm{ap}_{e_A}(y, x, p), e_B(y)(f(y)))}(e_A^{-1}(e_A(x)), \mathrm{ret}_{e_A}(x), x, \mathrm{refl}_A(x), \mathrm{ret}_{e_A}(x), \mathrm{apd}_{y:A.y =_A x}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x), \mathrm{ret}_{e_A}(x), \mathrm{refl}_A(x)) \\ \bullet \mathrm{ret}_{e_B}(x, f(x))):\mathrm{congform}(e_A, e_B)^{-1}(\mathrm{congform}(e_A, e_B)(f)) =_{\prod_{x:A} B(x)} f \end{array} </annotation></semantics></math></div> <p>We can define the witness that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathrm{congform}(e_A, e_B)^{-1}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/retraction">retraction</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{congform}(e_A, e_B)</annotation></semantics></math> as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center"><mtr><mtd><msub><mi mathvariant="normal">ret</mi> <mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>≔</mo></mtd></mtr> <mtr><mtd><msubsup><mi mathvariant="normal">ext</mi> <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> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><msub><mi mathvariant="normal">ap</mi> <mrow><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow></msub><mo stretchy="false">(</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo></mtd></mtr> <mtr><mtd><msub><mi mathvariant="normal">ap</mi> <mrow><mi>λ</mi><mi>p</mi><mo>:</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>A</mi><mo>′</mo></mrow></msub><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>.</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>p</mi><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ap</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">coh</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>•</mo><msub><mi mathvariant="normal">apbinary</mi> <mrow><mi>λ</mi><mi>y</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>λ</mi><mi>p</mi><mo>:</mo><mi>y</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi><mo>.</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ap</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">apd</mi> <mrow><mi>y</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>y</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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></mtd></mtr> <mtr><mtd><mo>•</mo><msub><mi mathvariant="normal">ret</mi> <mrow><msub><mi>e</mi> <mi>B</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><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>f</mi></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex"> \begin{array}{c} \mathrm{ret}_{\mathrm{congform}(e_A, e_B)}(f) \coloneqq \\ \mathrm{ext}_{\prod_{x:A} B(x)}^{-1}(\lambda x:A.\mathrm{ap}_{e_B(x)^{-1}}(\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{sec}_{e_A}(e_A(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x))))), e_B(x)(f(x)), \\ \mathrm{ap}_{\lambda p:e_A(e_A^{-1}(e_A(x))) =_{A'} e_A(x).\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), p, e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x)))))}(\mathrm{sec}_{e_A}(e_A(x)), \mathrm{ap}_{e_A}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x)), \mathrm{coh}_{e_A}(x) \\ \bullet \mathrm{apbinary}_{\lambda y:A.\lambda p:y =_A x.\mathrm{transport}_{x:A'.B'(x)}(e_A(y), e_A(x), \mathrm{ap}_{e_A}(y, x, p), e_B(y)(f(y)))}(e_A^{-1}(e_A(x)), \mathrm{ret}_{e_A}(x), x, \mathrm{refl}_A(x), \mathrm{ret}_{e_A}(x), \mathrm{apd}_{y:A.y =_A x}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x), \mathrm{ret}_{e_A}(x), \mathrm{refl}_A(x)) \\ \bullet \mathrm{ret}_{e_B}(x, f(x))):\mathrm{congform}(e_A, e_B)^{-1}(\mathrm{congform}(e_A, e_B)(f)) =_{\prod_{x:A} B(x)} f \end{array} </annotation></semantics></math></div><hr /> <p>Similarly, by definition,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{congform}(e_A, e_B)(\mathrm{congform}(e_A, e_B)^{-1}(f)) \equiv \lambda x:A'.\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), e_B(e_A^{-1}(x))(( \lambda x:A.e_B(x)^{-1}(f(e_A(x))) )(e_A^{-1}(x))))</annotation></semantics></math></div> <p>By the computation rules of strict dependent function types, there is a family of judgmental equalities</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mo stretchy="false">(</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A \vdash (\lambda x:A.e_B(x)^{-1}(f(e_A(x))))(e_A^{-1}(x)) \equiv e_B(e_A^{-1}(x))^{-1}(f(e_A(e_A^{-1}(x))))</annotation></semantics></math></div> <p>and thus by the structural rules of <a class="existingWikiWord" href="/nlab/show/judgmental+equalities">judgmental equalities</a> and the judgmental congruence rules for dependent function types, a judgmental equality</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lambda x:A'.\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), e_B(e_A^{-1}(x))((\lambda x:A.e_B(x)^{-1}(f(e_A(x))) )(e_A^{-1}(x)))) \equiv \lambda x:A'.\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(e_A(e_A^{-1}(x))))))</annotation></semantics></math></div> <p>From the family of elements</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>⊢</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A' \vdash \mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(e_A(e_A^{-1}(x))))))</annotation></semantics></math></div> <p>one could define the family of dependent functions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>⊢</mo><mi>λ</mi><mi>y</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>λ</mi><mi>p</mi><mo>:</mo><mi>y</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi><mo>.</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><mi>p</mi><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A' \vdash \lambda y:A.\lambda p:y =_A x.\mathrm{transport}_{x:A'.B'(x)}(y), x, p, e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(y))))</annotation></semantics></math></div> <p>This means by applying the above dependent function to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e_A(e_A^{-1}(x))</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{sec}_{e_A}(x)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">refl</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{refl}_A(x)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{sec}_{e_A}(x)</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">apd</mi> <mrow><mi>y</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>y</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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><annotation encoding="application/x-tex">\mathrm{apd}_{y:A.y =_A x}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), \mathrm{sec}_{e_A}(x), \mathrm{refl}_A(x))</annotation></semantics></math>, one gets the family of identifications</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>⊢</mo><msub><mi mathvariant="normal">apbinary</mi> <mrow><mi>λ</mi><mi>y</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>λ</mi><mi>p</mi><mo>:</mo><mi>y</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi><mo>.</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><mi>p</mi><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">apd</mi> <mrow><mi>y</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>y</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A' \vdash \mathrm{apbinary}_{\lambda y:A.\lambda p:y =_A x.\mathrm{transport}_{x:A'.B'(x)}(y), x, p, e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(y))))}(e_A(e_A^{-1}(x)), \mathrm{sec}_{e_A}(x), x, \mathrm{refl}_A(x), \mathrm{sec}_{e_A}(x), \mathrm{apd}_{y:A.y =_A x}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), \mathrm{sec}_{e_A}(x), \mathrm{refl}_A(x)))</annotation></semantics></math></div> <p>in type</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><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>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(e_A(e_A^{-1}(x)))))) =_{B'(x)} \mathrm{transport}_{x:A'.B'(x)}(x, x, \mathrm{refl}_A(x), e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(x))))</annotation></semantics></math></div> <p>By the judgmental computation rules of identity types, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><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>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{transport}_{x:A'.B'(x)}(x, x, \mathrm{refl}_A(x), e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(x)))) \equiv e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(x)))</annotation></semantics></math></div> <p>Similarly, The family of equivalences</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><msub><mi>e</mi> <mi>B</mi></msub><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><mo>≃</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A \vdash e_B(x):B(x) \simeq B'(e_A(x))</annotation></semantics></math></div> <p>has a family of identifications</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>,</mo><mi>y</mi><mo>:</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⊢</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>B</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>:</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><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>y</mi></mrow><annotation encoding="application/x-tex">x:A, y:B'(e_A(x)) \vdash \mathrm{sec}_{e_B}(x, y):e_B(x)(e_B(x)^{-1}(y)) =_{B(x)} y</annotation></semantics></math></div> <p>witnessing that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">e_B(x)^{-1}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/section">section</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e_B(x)</annotation></semantics></math> for each <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>.</p> <p>By substituting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(e_A(e_A^{-1}(x)))</annotation></semantics></math> in for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e_A^{-1}(x)</annotation></semantics></math> in for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> in the above expression we get the family of identifications</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>⊢</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>B</mi></msub></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <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>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A' \vdash \mathrm{sec}_{e_B}(e_A^{-1}(x), f(e_A(e_A^{-1}(x)))):e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(e_A(e_A^{-1}(x))))) =_{B'(e_A(e_A^{-1}(x)))} f(e_A(e_A^{-1}(x)))</annotation></semantics></math></div> <p>and by transport across <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{sec}_{e_A}(x)</annotation></semantics></math> we get</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>⊢</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>z</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></msub><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>B</mi></msub></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <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>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A' \vdash \mathrm{transport}_{z:A'.e_B(x)(e_B(x)^{-1}(f(z))) =_{B'(z)} f(z)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), \mathrm{sec}_{e_B}(e_A^{-1}(x), f(e_A(e_A^{-1}(x))))):e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(x))) =_{B'(x)} f(x)</annotation></semantics></math></div> <p>By concatenation of identifications we get</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center"><mtr><mtd><msub><mi mathvariant="normal">apbinary</mi> <mrow><mi>λ</mi><mi>y</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>λ</mi><mi>p</mi><mo>:</mo><mi>y</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi><mo>.</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><mi>p</mi><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">apd</mi> <mrow><mi>y</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>y</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>•</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>z</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></msub><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>B</mi></msub></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <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>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex"> \begin{array}{c} \mathrm{apbinary}_{\lambda y:A.\lambda p:y =_A x.\mathrm{transport}_{x:A'.B'(x)}(y), x, p, e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(y))))}(e_A(e_A^{-1}(x)), \mathrm{sec}_{e_A}(x), x, \mathrm{refl}_A(x), \mathrm{sec}_{e_A}(x), \mathrm{apd}_{y:A.y =_A x}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), \mathrm{sec}_{e_A}(x), \mathrm{refl}_A(x))) \\ \bullet \mathrm{transport}_{z:A'.e_B(x)(e_B(x)^{-1}(f(z))) =_{B'(z)} f(z)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), \mathrm{sec}_{e_B}(e_A^{-1}(x), f(e_A(e_A^{-1}(x))))) \end{array} </annotation></semantics></math></div> <p>in type</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(e_A(e_A^{-1}(x)))))) =_{B'(x)} f(x)</annotation></semantics></math></div> <p>and by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math>-abstraction we get</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center"><mtr><mtd><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><msub><mi mathvariant="normal">apbinary</mi> <mrow><mi>λ</mi><mi>y</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>λ</mi><mi>p</mi><mo>:</mo><mi>y</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi><mo>.</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><mi>p</mi><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">apd</mi> <mrow><mi>y</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>y</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>•</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>z</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></msub><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>B</mi></msub></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <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>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex"> \begin{array}{c} \lambda x:A'.\mathrm{apbinary}_{\lambda y:A.\lambda p:y =_A x.\mathrm{transport}_{x:A'.B'(x)}(y), x, p, e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(y))))}(e_A(e_A^{-1}(x)), \mathrm{sec}_{e_A}(x), x, \mathrm{refl}_A(x), \mathrm{sec}_{e_A}(x), \mathrm{apd}_{y:A.y =_A x}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), \mathrm{sec}_{e_A}(x), \mathrm{refl}_A(x))) \\ \bullet \mathrm{transport}_{z:A'.e_B(x)(e_B(x)^{-1}(f(z))) =_{B'(z)} f(z)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), \mathrm{sec}_{e_B}(e_A^{-1}(x), f(e_A(e_A^{-1}(x))))) \end{array} </annotation></semantics></math></div> <p>in type</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo></mrow></munder><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\prod_{x:A'} \mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(e_A(e_A^{-1}(x)))))) =_{B'(x)} f(x)</annotation></semantics></math></div> <p>By dependent function extensionality, we get the identification</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center"><mtr><mtd><msubsup><mi mathvariant="normal">ext</mi> <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> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo></mtd></mtr> <mtr><mtd><msub><mi mathvariant="normal">apbinary</mi> <mrow><mi>λ</mi><mi>y</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>λ</mi><mi>p</mi><mo>:</mo><mi>y</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi><mo>.</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><mi>p</mi><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">apd</mi> <mrow><mi>y</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>y</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>•</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>z</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></msub><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>B</mi></msub></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <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>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex"> \begin{array}{c} \mathrm{ext}_{\prod_{x:A} B(x)}^{-1}(\lambda x:A'. \\ \mathrm{apbinary}_{\lambda y:A.\lambda p:y =_A x.\mathrm{transport}_{x:A'.B'(x)}(y), x, p, e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(y))))}(e_A(e_A^{-1}(x)), \mathrm{sec}_{e_A}(x), x, \mathrm{refl}_A(x), \mathrm{sec}_{e_A}(x), \mathrm{apd}_{y:A.y =_A x}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), \mathrm{sec}_{e_A}(x), \mathrm{refl}_A(x))) \\ \bullet \mathrm{transport}_{z:A'.e_B(x)(e_B(x)^{-1}(f(z))) =_{B'(z)} f(z)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), \mathrm{sec}_{e_B}(e_A^{-1}(x), f(e_A(e_A^{-1}(x)))))) \end{array} </annotation></semantics></math></div> <p>in type</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><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></mrow></msub><mi>f</mi></mrow><annotation encoding="application/x-tex">\lambda x:A'.\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(e_A(e_A^{-1}(x)))))) =_{\prod_{x:A'} B'(x)} f</annotation></semantics></math></div> <p>and since we defined</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{congform}(e_A, e_B)(\mathrm{congform}(e_A, e_B)^{-1}(f)) \equiv \lambda x:A'.\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), e_B(e_A^{-1}(x))(( \lambda x:A.e_B(x)^{-1}(f(e_A(x))) )(e_A^{-1}(x))))</annotation></semantics></math></div> <p>we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center"><mtr><mtd><msubsup><mi mathvariant="normal">ext</mi> <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> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo></mtd></mtr> <mtr><mtd><msub><mi mathvariant="normal">apbinary</mi> <mrow><mi>λ</mi><mi>y</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>λ</mi><mi>p</mi><mo>:</mo><mi>y</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi><mo>.</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><mi>p</mi><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">apd</mi> <mrow><mi>y</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>y</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>•</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>z</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></msub><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>B</mi></msub></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <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>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><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></mrow></msub><mi>f</mi></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex"> \begin{array}{c} \mathrm{ext}_{\prod_{x:A} B(x)}^{-1}(\lambda x:A'. \\ \mathrm{apbinary}_{\lambda y:A.\lambda p:y =_A x.\mathrm{transport}_{x:A'.B'(x)}(y), x, p, e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(y))))}(e_A(e_A^{-1}(x)), \mathrm{sec}_{e_A}(x), x, \mathrm{refl}_A(x), \mathrm{sec}_{e_A}(x), \mathrm{apd}_{y:A.y =_A x}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), \mathrm{sec}_{e_A}(x), \mathrm{refl}_A(x))) \\ \bullet \mathrm{transport}_{z:A'.e_B(x)(e_B(x)^{-1}(f(z))) =_{B'(z)} f(z)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), \mathrm{sec}_{e_B}(e_A^{-1}(x), f(e_A(e_A^{-1}(x)))))):\mathrm{congform}(e_A, e_B)(\mathrm{congform}(e_A, e_B)^{-1}(f)) =_{\prod_{x:A'} B'(x)} f \end{array} </annotation></semantics></math></div> <p>We can define the witness that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathrm{congform}(e_A, e_B)^{-1}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/section">section</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{congform}(e_A, e_B)</annotation></semantics></math> as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center"><mtr><mtd><msub><mi mathvariant="normal">sec</mi> <mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>≔</mo><msubsup><mi mathvariant="normal">ext</mi> <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> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo></mtd></mtr> <mtr><mtd><msub><mi mathvariant="normal">apbinary</mi> <mrow><mi>λ</mi><mi>y</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>λ</mi><mi>p</mi><mo>:</mo><mi>y</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi><mo>.</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><mi>p</mi><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">apd</mi> <mrow><mi>y</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>y</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><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><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>•</mo><msub><mi mathvariant="normal">transport</mi> <mrow><mi>z</mi><mo>:</mo><mi>A</mi><mo>′</mo><mo>.</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></msub><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">sec</mi> <mrow><msub><mi>e</mi> <mi>B</mi></msub></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <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>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><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></mrow></msub><mi>f</mi></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex"> \begin{array}{c} \mathrm{sec}_{\mathrm{congform}(e_A, e_B)}(f) \coloneqq \mathrm{ext}_{\prod_{x:A} B(x)}^{-1}(\lambda x:A'. \\ \mathrm{apbinary}_{\lambda y:A.\lambda p:y =_A x.\mathrm{transport}_{x:A'.B'(x)}(y), x, p, e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(y))))}(e_A(e_A^{-1}(x)), \mathrm{sec}_{e_A}(x), x, \mathrm{refl}_A(x), \mathrm{sec}_{e_A}(x), \mathrm{apd}_{y:A.y =_A x}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), \mathrm{sec}_{e_A}(x), \mathrm{refl}_A(x))) \\ \bullet \mathrm{transport}_{z:A'.e_B(x)(e_B(x)^{-1}(f(z))) =_{B'(z)} f(z)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), \mathrm{sec}_{e_B}(e_A^{-1}(x), f(e_A(e_A^{-1}(x)))))):\mathrm{congform}(e_A, e_B)(\mathrm{congform}(e_A, e_B)^{-1}(f)) =_{\prod_{x:A'} B'(x)} f \end{array} </annotation></semantics></math></div><hr /> <p>Since we have functions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo>:</mo><mrow><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></mrow><mo>→</mo><mrow><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><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\mathrm{congform}(e_A, e_B):\left(\prod_{x:A} B(x)\right) \to \left(\prod_{x:A'} B'(x)\right)</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>:</mo><mrow><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><mo>)</mo></mrow><mo>→</mo><mrow><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></mrow></mrow><annotation encoding="application/x-tex">\mathrm{congform}(e_A, e_B)^{-1}:\left(\prod_{x:A'} B'(x)\right) \to \left(\prod_{x:A} B(x)\right)</annotation></semantics></math></div> <p>and families of identifications</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><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><mo>⊢</mo><msub><mi mathvariant="normal">ret</mi> <mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>:</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><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>f</mi></mrow><annotation encoding="application/x-tex">f:\prod_{x:A} B(x) \vdash \mathrm{ret}_{\mathrm{congform}(e_A, e_B)}(f):\mathrm{congform}(e_A, e_B)^{-1}(\mathrm{congform}(e_A, e_B)(f)) =_{\prod_{x:A} B(x)} f</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>g</mi><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><mo>⊢</mo><msub><mi mathvariant="normal">sec</mi> <mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>:</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><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></mrow></msub><mi>g</mi></mrow><annotation encoding="application/x-tex">g:\prod_{x:A'} B'(x) \vdash \mathrm{sec}_{\mathrm{congform}(e_A, e_B)}(g):\mathrm{congform}(e_A, e_B)(\mathrm{congform}(e_A, e_B)^{-1}(g)) =_{\prod_{x:A'} B'(x)} g</annotation></semantics></math></div> <p>we could form the equivalence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mi mathvariant="normal">qInvToIsEquiv</mi><mrow><mo>(</mo><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>,</mo><mi>λ</mi><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><mo>.</mo><msub><mi mathvariant="normal">ret</mi> <mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>,</mo><mi>λ</mi><mi>g</mi><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><mo>.</mo><msub><mi mathvariant="normal">sec</mi> <mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>)</mo></mrow><mo>:</mo><mrow><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></mrow><mo>≃</mo><mrow><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><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\left(\mathrm{congform}(e_A, e_B), \mathrm{qInvToIsEquiv}\left(\mathrm{congform}(e_A, e_B)^{-1}, \lambda f:\prod_{x:A} B(x).\mathrm{ret}_{\mathrm{congform}(e_A, e_B)}(f), \lambda g:\prod_{x:A'} B'(x).\mathrm{sec}_{\mathrm{congform}(e_A, e_B)}(g)\right)\right):\left(\prod_{x:A} B(x)\right) \simeq \left(\prod_{x:A'} B'(x)\right)</annotation></semantics></math></div> <p>By a common abuse of notation we denote the equivalence by the same name as the underlying function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{congform}(e_A, e_B)</annotation></semantics></math>; thus we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo>:</mo><mrow><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></mrow><mo>≃</mo><mrow><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><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\mathrm{congform}(e_A, e_B):\left(\prod_{x:A} B(x)\right) \simeq \left(\prod_{x:A'} B'(x)\right)</annotation></semantics></math></div> <p></p> </div> </p> <h5 id="using_identity_types_between_types">Using identity types between types</h5> <p>We assume a <a class="existingWikiWord" href="/nlab/show/dependent+type+theory+with+type+variables">dependent type theory with type variables</a> and <a class="existingWikiWord" href="/nlab/show/identity+type#IdentityTypesBetweenTypes">identity types between types</a>.</p> <p> <div class='num_theorem'> <h6>Theorem</h6> <p>Given types <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> and type families <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>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A \vdash B(x)</annotation></semantics></math>, <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>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A' \vdash B'(x)</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/identity+type#IdentityTypesBetweenTypes">identity types between types</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>A</mi></msub><mo>:</mo><mi>A</mi><mo>=</mo><mi>A</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">p_A:A = A'</annotation></semantics></math> and dependent function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>B</mi></msub><mo>:</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><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 mathvariant="normal">idtoequiv</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>A</mi></msub><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p_B:\prod_{x:A} B(x) = B'(\mathrm{idtoequiv}(p_A, x))</annotation></semantics></math> consisting of a family of identity types between types, there is an identity type between the types</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>p</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo>:</mo><mrow><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></mrow><mo>=</mo><mrow><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><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\mathrm{congform}(p_A, p_B):\left(\prod_{x:A} B(x)\right) = \left(\prod_{x:A'} B'(x)\right)</annotation></semantics></math></div> <p></p> </div> </p> <p>…</p> <h4 id="weak_dependent_product_types">Weak dependent product types</h4> <p> <div class='num_theorem'> <h6>Theorem</h6> <p>Assuming dependent function extensionality, given types <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> and type families <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>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A \vdash B(x)</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>B</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A \vdash B'(x)</annotation></semantics></math> and equivalences <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>A</mi></msub><mo>:</mo><mi>A</mi><mo>≃</mo><mi>A</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">e_A:A \simeq A'</annotation></semantics></math> and dependent function of equivalences <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>B</mi></msub><mo>:</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><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>e</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e_B:\prod_{x:A} B(x) \simeq B'(e(x))</annotation></semantics></math>, there is an equivalence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">congform</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo>:</mo><mrow><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></mrow><mo>≃</mo><mrow><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><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\mathrm{congform}(e_A, e_B):\left(\prod_{x:A} B(x)\right) \simeq \left(\prod_{x:A'} B'(x)\right)</annotation></semantics></math></div> <p></p> </div> </p> <p>Since <a class="existingWikiWord" href="/nlab/show/dependent+function+types">dependent function types</a> are negative types, we first present the typal congruence rule for the elimination rule of dependent function types</p> <p> <div class='num_theorem'> <h6>Theorem</h6> <p>Given a 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> and a type family <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>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A \vdash B(x)</annotation></semantics></math>, dependent functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f:\prod_{x:A} B(x)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g:\prod_{x:A} B(x)</annotation></semantics></math> and an identification <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>f</mi><msub><mo>=</mo> <mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mi>g</mi></mrow><annotation encoding="application/x-tex">p:f =_{\prod_{x:A} B(x)} g</annotation></semantics></math> there are families of identifications <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 mathvariant="normal">compelim</mi><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><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><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><annotation encoding="application/x-tex">x:A \vdash \mathrm{compelim}(f, g, p)(x):f(x) =_{B(x)} g(x)</annotation></semantics></math>.</p> </div> </p> <p> <div class='proof'> <h6>Proof</h6> <p>We simply define the dependent function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">compelim</mi></mrow><annotation encoding="application/x-tex">\mathrm{compelim}</annotation></semantics></math> to be happly, which is inductively defined on <a class="existingWikiWord" href="/nlab/show/identity+types">identity types</a>.</p> </div> </p> <p>The next is the typal congruence rule for the uniqueness rule of dependent function types.</p> <p> <div class='num_theorem'> <h6>Theorem</h6> <p>For weak dependent product types with dependent function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><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><annotation encoding="application/x-tex">\eta:\prod_{f:\prod_{x:A} B(x)} f =_{\prod_{x:A} B(x)} \lambda x:A.f(x)</annotation></semantics></math></div> <p>given</p> <ul> <li> <p>a 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></p> </li> <li> <p>a type family <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>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A \vdash B(x)</annotation></semantics></math></p> </li> <li> <p>dependent functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f:\prod_{x:A} B(x)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>′</mo><mo>:</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f':\prod_{x:A} B(x)</annotation></semantics></math></p> </li> <li> <p>an identification <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>f</mi><msub><mo>=</mo> <mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mi>f</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">p:f =_{\prod_{x:A} B(x)} f'</annotation></semantics></math>,</p> </li> </ul> <p>there is a family of identifications</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">etaCong</mi> <mo lspace="thinmathspace" rspace="thinmathspace">∏</mo></msub><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>f</mi><mo>′</mo><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo>:</mo><mi mathvariant="normal">transport</mi><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>f</mi><mo>′</mo><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>η</mi> <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><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><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></msub><msub><mi>η</mi> <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><mo stretchy="false">(</mo><mi>f</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{etaCong}_{\prod}(f, f', p):\mathrm{transport}(f, f', p)(\eta_{\prod_{x:A} B(x)}(f)) =_{f =_{\prod_{x:A} B(x)} \lambda x:A.f(x)} \eta_{\prod_{x:A} B(x)}(f')</annotation></semantics></math></div> <p></p> </div> </p> <p> <div class='proof'> <h6>Proof</h6> <p>We simply define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">etaCong</mi> <mo lspace="thinmathspace" rspace="thinmathspace">∏</mo></msub><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>f</mi><mo>′</mo><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{etaCong}_{\prod}(f, f', p)</annotation></semantics></math> to be the <a class="existingWikiWord" href="/nlab/show/dependent+function+application+to+identifications">dependent function application to identifications</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">apd</mi><mo stretchy="false">(</mo><mi>η</mi><mo>,</mo><mi>f</mi><mo>,</mo><mi>f</mi><mo>′</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{apd}(\eta, f, f' p)</annotation></semantics></math>.</p> </div> </p> <p>The next is the typal congruence rule for the introduction rule of function types. However, unlike the case for the other two rules, one needs dependent function extensionality.</p> <p> <div class='num_theorem'> <h6>Theorem</h6> <p>Assuming dependent function extensionality, given a 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> and a type family <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>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A \vdash B(x)</annotation></semantics></math>, families of elements <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>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></mrow><annotation encoding="application/x-tex">x:A \vdash b(x):B(x)</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>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></mrow><annotation encoding="application/x-tex">x:A \vdash b'(x):B(x)</annotation></semantics></math>, and families of <a class="existingWikiWord" href="/nlab/show/identifications">identifications</a> <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>p</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><msub><mo>=</mo> <mrow><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mi>b</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A \vdash p(x):b(x) =_{B(x)} b'(x)</annotation></semantics></math>, there is an identification</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">congintro</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><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><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><mo stretchy="false">(</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo stretchy="false">)</mo><mo>.</mo><mi>b</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{congintro}_{x:A.p(x)}:\lambda (x:A).b(x) =_{\prod_{x:A} B(x)} \lambda (x:A).b'(x)</annotation></semantics></math></div> <p></p> </div> </p> <p> <div class='proof'> <h6>Proof</h6> <p>By the computation rule of weak dependent function types, there are families of identifications</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><msubsup><mi>β</mi> <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> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>b</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</mo><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><msub><mo>=</mo> <mrow><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mi>b</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A \vdash \beta_{\prod_{x:A} B(x)}^{x:A.b(x)}(b(x)):((\lambda x:A.b(x))(x) =_{B(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><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><msubsup><mi>β</mi> <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> <mrow><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></mrow></msubsup><mo stretchy="false">(</mo><mi>b</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</mo><mo stretchy="false">(</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 stretchy="false">)</mo><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>b</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A \vdash \beta_{\prod_{x:A} B(x)}^{x:A.b'(x)}(b'(x)):((\lambda x:A.b'(x))(x) =_{B(x)} b'(x)</annotation></semantics></math></div> <p>Thus, there are families of identificaitons</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><msubsup><mi>β</mi> <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> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>b</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>•</mo><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>•</mo><msubsup><mi>β</mi> <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> <mrow><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></mrow></msubsup><mo stretchy="false">(</mo><mi>b</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>:</mo><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><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>′</mo><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><annotation encoding="application/x-tex">x:A \vdash \beta_{\prod_{x:A} B(x)}^{x:A.b(x)}(b(x)) \bullet p(x) \bullet \beta_{\prod_{x:A} B(x)}^{x:A.b'(x)}(b'(x))^{-1}:(\lambda x:A.b(x))(x) =_{B(x)} (\lambda x:A.b'(x))(x)</annotation></semantics></math></div> <p>and by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math>-abstraction, one gets the dependent function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo stretchy="false">)</mo><mo>.</mo><msubsup><mi>β</mi> <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> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>b</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>•</mo><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>•</mo><msubsup><mi>β</mi> <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> <mrow><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></mrow></msubsup><mo stretchy="false">(</mo><mi>b</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>:</mo><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><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>′</mo><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><annotation encoding="application/x-tex">\lambda (x:A).\beta_{\prod_{x:A} B(x)}^{x:A.b(x)}(b(x)) \bullet p(x) \bullet \beta_{\prod_{x:A} B(x)}^{x:A.b'(x)}(b'(x))^{-1}:(\lambda x:A.b(x))(x) =_{B(x)} (\lambda x:A.b'(x))(x)</annotation></semantics></math></div> <p>By dependent function extensionality, there is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+types">equivalence of types</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi mathvariant="normal">ext</mi> <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> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>:</mo><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><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><mo stretchy="false">(</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 stretchy="false">)</mo><mo>≃</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><msub><mi mathvariant="normal">Id</mi> <mrow><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><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><mo>,</mo><mo stretchy="false">(</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 stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{ext}_{\prod_{x:A} B(x)}^{-1}:(\lambda x:A.b(x)) =_{\prod_{x:A} B(x)} (\lambda x:A.b'(x)) \simeq \prod_{x:A} \mathrm{Id}_{B(x)}((\lambda x:A.b(x))(x), (\lambda x:A.b'(x))(x))</annotation></semantics></math></div> <p>which yields an identification</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi mathvariant="normal">ext</mi> <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> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><msup><mo></mo><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo stretchy="false">)</mo><mo>.</mo><msubsup><mi>β</mi> <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> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>b</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>•</mo><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>•</mo><msubsup><mi>β</mi> <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> <mrow><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></mrow></msubsup><mo stretchy="false">(</mo><mi>b</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>:</mo><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><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><mo stretchy="false">(</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 stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{ext}_{\prod_{x:A} B(x)}^{-1}^{-1}(\lambda (x:A).\beta_{\prod_{x:A} B(x)}^{x:A.b(x)}(b(x)) \bullet p(x) \bullet \beta_{\prod_{x:A} B(x)}^{x:A.b'(x)}(b'(x))^{-1}):(\lambda x:A.b(x)) =_{\prod_{x:A} B(x)} (\lambda x:A.b'(x))</annotation></semantics></math></div> <p>We define</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">congintro</mi> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo>≔</mo><msubsup><mi mathvariant="normal">ext</mi> <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> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><msup><mo></mo><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo stretchy="false">)</mo><mo>.</mo><msubsup><mi>β</mi> <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> <mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>b</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>•</mo><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>•</mo><msubsup><mi>β</mi> <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> <mrow><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></mrow></msubsup><mo stretchy="false">(</mo><mi>b</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>:</mo><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><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><mo stretchy="false">(</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 stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{congintro}_{x:A.p(x)} \coloneqq \mathrm{ext}_{\prod_{x:A} B(x)}^{-1}^{-1}(\lambda (x:A).\beta_{\prod_{x:A} B(x)}^{x:A.b(x)}(b(x)) \bullet p(x) \bullet \beta_{\prod_{x:A} B(x)}^{x:A.b'(x)}(b'(x))^{-1}):(\lambda x:A.b(x)) =_{\prod_{x:A} B(x)} (\lambda x:A.b'(x))</annotation></semantics></math></div> <p></p> </div> </p> <h3 id="application_in_logic">Application in logic</h3> <p>In <a class="existingWikiWord" href="/nlab/show/logic">logic</a>, dependent functions types express <a class="existingWikiWord" href="/nlab/show/universal+quantifications">universal quantifications</a>. More precisely, for <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>ϕ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A \vdash \phi(x)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/predicate">predicate</a> on a 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>, under <a class="existingWikiWord" href="/nlab/show/propositions+as+types">propositions as types</a> the <a class="existingWikiWord" href="/nlab/show/universal+quantification">universal quantification</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∀</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\forall x:A.\phi(x)</annotation></semantics></math> is the dependent product type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>ϕ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\prod_{x:A} \phi(x)</annotation></semantics></math> (or rather the <a class="existingWikiWord" href="/nlab/show/bracket+type">bracket type</a> of that if one wishes to force this to be of type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Prop</mi></mrow><annotation encoding="application/x-tex">Prop</annotation></semantics></math> again ).</p> <h3 id="graph_of_a_dependent_function">Graph of a dependent function</h3> <p>Given a 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> and a type family <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>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A \vdash B(x)</annotation></semantics></math>, there is a function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">graph</mi><mo>:</mo><mrow><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></mrow><mo>→</mo><mrow><mo>(</mo><mi>A</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></mrow></mrow><annotation encoding="application/x-tex">\mathrm{graph}:\left(\prod_{x:A} B(x)\right) \to \left(A \to \sum_{x:A} B(x)\right)</annotation></semantics></math></div> <p>which takes a dependent function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f:\prod_{x:A} B(x)</annotation></semantics></math> and returns the <strong><a class="existingWikiWord" href="/nlab/show/graph+of+a+function">graph of a dependent function</a></strong></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">graph</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>:</mo><mi>A</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><annotation encoding="application/x-tex">\mathrm{graph}(f):A \to \sum_{x:A} B(x)</annotation></semantics></math></div> <p>defined by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">graph</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≡</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{graph}(f)(x) \equiv (x, f(x))</annotation></semantics></math> for all <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>. As a <a class="existingWikiWord" href="/nlab/show/dependent+anafunction">dependent anafunction</a> the graph of the dependent function is represented by the <a class="existingWikiWord" href="/nlab/show/identity+type">identity type</a> family</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>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>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>y</mi></mrow><annotation encoding="application/x-tex">x:A, y:B(x) \vdash f(x) =_{B(x)} y</annotation></semantics></math></div> <h3 id="relation_to_sections">Relation to sections</h3> <p>A family of type <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>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A \vdash B(x)</annotation></semantics></math> is equivalently a type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> with a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>B</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">f:B \to A</annotation></semantics></math>. Then each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(x)</annotation></semantics></math> is defined as the fiber of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> at element <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>. Then the dependent product of a function is defined as the dependent product type</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>≔</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>y</mi><mo>:</mo><mi>B</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi></mrow><annotation encoding="application/x-tex">\Pi_{A, B}(f) \coloneqq \prod_{x:A} \sum_{y:B} f(y) =_A x</annotation></semantics></math></div> <p>or equivalently, due to the <a class="existingWikiWord" href="/nlab/show/type+theoretic+axiom+of+choice">type theoretic axiom of choice</a>, as the dependent sum type</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>≔</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>g</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow></munder><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>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi></mrow><annotation encoding="application/x-tex">\Pi_{A, B}(f) \coloneqq \sum_{g:A \to B} \prod_{x:A} f(g(x)) =_A x</annotation></semantics></math></div> <p>which says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/section">section</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>. One could eliminate the use of the dependent product type entirely by using the definition of dependent product type from function types:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>≔</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>g</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>h</mi><mo>:</mo><mi>A</mi><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>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>A</mi></msub><mi>x</mi></mrow></munder><msubsup><mi mathvariant="normal">pr</mi> <mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>A</mi></msubsup><mo>∘</mo><mi>h</mi><msub><mo>=</mo> <mrow><mi>A</mi><mo>→</mo><mi>A</mi></mrow></msub><msub><mi mathvariant="normal">id</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\Pi_{A, B}(f) \coloneqq \sum_{g:A \to B} \sum_{h:A \to \sum_{x:A} f(g(x)) =_A x} \mathrm{pr}_{\sum}^{A} \circ h =_{A \to A} \mathrm{id}_A</annotation></semantics></math></div> <h2 id="categorical_interpretation">Categorical interpretation</h2> <p>In <a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a>, the <a class="existingWikiWord" href="/nlab/show/dependent+product+types">dependent product types</a> are <em><a class="existingWikiWord" href="/nlab/show/relative+adjoint+functor">relative</a></em> <a class="existingWikiWord" href="/nlab/show/right+adjoints">right adjoints</a> to <a class="existingWikiWord" href="/nlab/show/context+extension">context extension</a> in <a class="existingWikiWord" href="/nlab/show/comprehension+categories">comprehension categories</a>.</p> <p>There is also another interpretation in category theory of the dependent product type over <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>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><annotation encoding="application/x-tex">x:A \vdash B(x) \; \mathrm{type}</annotation></semantics></math> as the <a class="existingWikiWord" href="/nlab/show/terminal">terminal</a> <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>-indexed <a class="existingWikiWord" href="/nlab/show/wide+span">wide span</a> under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(x)</annotation></semantics></math>, the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\prod_{x:A} B(x)</annotation></semantics></math> with a family of morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><mrow><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></mrow><mo>→</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_A(x):\left(\prod_{x:A} B(x)\right) \to B(x)</annotation></semantics></math></div> <p>such that for any other object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> with a family of morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><mi>C</mi><mo>→</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x):C \to B(x)</annotation></semantics></math>, there exists a unique morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>u</mi> <mi>C</mi></msub><mo>:</mo><mi>C</mi><mo>→</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u_C:C \to \prod_{x:A} B(x)</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><msub><mi>π</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>u</mi> <mi>C</mi></msub><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_A(x) \circ u_C = f(x)</annotation></semantics></math></div> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a></li> <li><a class="existingWikiWord" href="/nlab/show/dependent+sum+type">dependent sum type</a></li> <li><a class="existingWikiWord" href="/nlab/show/dependent+pullback+type">dependent pullback type</a></li> <li><a class="existingWikiWord" href="/nlab/show/function+type">function type</a></li> <li><a class="existingWikiWord" href="/nlab/show/dependent+sequence+type">dependent sequence type</a></li> <li><a class="existingWikiWord" href="/nlab/show/dependent+extension+type">dependent extension type</a></li> </ul> <h2 id="references">References</h2> <p>The standard rules for type-formation, term introduction/elimination and computation of dependent product type are listed for instance in <a href="http://www.math.unipa.it/~ngambino/Research/Lectures/lecture1.pdf">part I</a> of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Nicola+Gambino">Nicola Gambino</a>, <em>Lectures on dependent type theory</em> (<a href="http://www.cs.le.ac.uk/events/mgs2009/courses/gambino/lecturenotes-gambino.pdf">pdf</a>)</li> </ul> <p>Another textbook account could be found in section 2.1 of:</p> <ul> <li id="Rijke22"><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://arxiv.org/abs/2212.11082">arXiv:2212.11082</a>)</li> </ul> <p>as well as sections 1.4 and 2.9 of:</p> <ul> <li id="UFP13"><a class="existingWikiWord" href="/nlab/show/Univalent+Foundations+Project">Univalent Foundations Project</a>, <em><a class="existingWikiWord" href="/nlab/show/HoTT+book">Homotopy Type Theory – Univalent Foundations of Mathematics</a></em> (2013)</li> </ul> <p>See also:</p> <ul> <li id="GarnerSDP"><a class="existingWikiWord" href="/nlab/show/Richard+Garner">Richard Garner</a>, <em>On the strength of dependent products in the type theory of Martin-Löf</em>, <a href="http://arxiv.org/abs/0803.4466">arXiv</a>.</li> </ul> <p>On the <a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a> of <a class="existingWikiWord" href="/nlab/show/dependent+product+types">dependent product types</a> as <em><a class="existingWikiWord" href="/nlab/show/relative+adjoint+functor">relative</a></em> <a class="existingWikiWord" href="/nlab/show/right+adjoints">right adjoints</a> to <a class="existingWikiWord" href="/nlab/show/context+extension">context extension</a> in <a class="existingWikiWord" href="/nlab/show/comprehension+categories">comprehension categories</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Michael+Lindgren">Michael Lindgren</a>, <em>Dependent products as relative adjoints</em>, Stockholm (2021) [<a class="existingWikiWord" href="/nlab/files/Lindgren-DependentProductsAsRelativeAdjoints.pdf" title="pdf">pdf</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 17, 2024 at 16:02:06. 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