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1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> natural numbers type </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/4584/#Item_10" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="type_theory">Type theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/natural+deduction">natural deduction</a></strong> <a class="existingWikiWord" href="/nlab/show/metalanguage">metalanguage</a>, <a class="existingWikiWord" href="/nlab/show/practical+foundations">practical foundations</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/judgement">judgement</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypothetical+judgement">hypothetical judgement</a>, <a class="existingWikiWord" href="/nlab/show/sequent">sequent</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/antecedents">antecedents</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊢</mo></mrow><annotation encoding="application/x-tex">\vdash</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/consequent">consequent</a>, <a class="existingWikiWord" href="/nlab/show/succedents">succedents</a></li> </ul> </li> </ul> <ol> <li><a class="existingWikiWord" href="/nlab/show/type+formation+rule">type formation rule</a></li> <li><a class="existingWikiWord" href="/nlab/show/term+introduction+rule">term introduction rule</a></li> <li><a class="existingWikiWord" href="/nlab/show/term+elimination+rule">term elimination rule</a></li> <li><a class="existingWikiWord" href="/nlab/show/computation+rule">computation rule</a></li> </ol> <p><strong><a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a></strong> (<a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent</a>, <a class="existingWikiWord" href="/nlab/show/intensional+type+theory">intensional</a>, <a class="existingWikiWord" href="/nlab/show/observational+type+theory">observational type theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>)</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/calculus+of+constructions">calculus of constructions</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/syntax">syntax</a></strong> <a class="existingWikiWord" href="/nlab/show/object+language">object language</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/theory">theory</a>, <a class="existingWikiWord" href="/nlab/show/axiom">axiom</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proposition">proposition</a>/<a class="existingWikiWord" href="/nlab/show/type">type</a> (<a class="existingWikiWord" href="/nlab/show/propositions+as+types">propositions as types</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/definition">definition</a>/<a class="existingWikiWord" href="/nlab/show/proof">proof</a>/<a class="existingWikiWord" href="/nlab/show/program">program</a> (<a class="existingWikiWord" href="/nlab/show/proofs+as+programs">proofs as programs</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/theorem">theorem</a></p> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/computational+trinitarianism">computational trinitarianism</a></strong> = <br /> <strong><a class="existingWikiWord" href="/nlab/show/propositions+as+types">propositions as types</a></strong> +<strong><a class="existingWikiWord" href="/nlab/show/programs+as+proofs">programs as proofs</a></strong> +<strong><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation type theory/category theory</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/logic">logic</a></th><th><a class="existingWikiWord" href="/nlab/show/set+theory">set theory</a> (<a class="existingWikiWord" href="/nlab/show/internal+logic+of+set+theory">internal logic</a> of)</th><th><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></th><th><a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/proposition">proposition</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/set">set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/object">object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type">type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/predicate">predicate</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/family+of+sets">family of sets</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/display+morphism">display morphism</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+type">dependent type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/proof">proof</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/element">element</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/generalized+element">generalized element</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/term">term</a>/<a class="existingWikiWord" href="/nlab/show/program">program</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cut+rule">cut rule</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/composition">composition</a> of <a class="existingWikiWord" href="/nlab/show/classifying+morphisms">classifying morphisms</a> / <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of <a class="existingWikiWord" href="/nlab/show/display+maps">display maps</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/substitution">substitution</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/introduction+rule">introduction rule</a> for <a class="existingWikiWord" href="/nlab/show/implication">implication</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/counit">counit</a> for hom-tensor adjunction</td><td style="text-align: left;">lambda</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/elimination+rule">elimination rule</a> for <a class="existingWikiWord" href="/nlab/show/implication">implication</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/unit">unit</a> for hom-tensor adjunction</td><td style="text-align: left;">application</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cut+elimination">cut elimination</a> for <a class="existingWikiWord" href="/nlab/show/implication">implication</a></td><td style="text-align: left;"></td><td style="text-align: left;">one of the <a class="existingWikiWord" href="/nlab/show/zigzag+identities">zigzag identities</a> for hom-tensor adjunction</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/beta+reduction">beta reduction</a></td></tr> <tr><td style="text-align: left;">identity elimination for <a class="existingWikiWord" href="/nlab/show/implication">implication</a></td><td style="text-align: left;"></td><td style="text-align: left;">the other <a class="existingWikiWord" href="/nlab/show/zigzag+identity">zigzag identity</a> for hom-tensor adjunction</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/eta+conversion">eta conversion</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/true">true</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/singleton">singleton</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>/<a class="existingWikiWord" href="/nlab/show/%28-2%29-truncated+object">(-2)-truncated object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/h-level+0">h-level 0</a>-<a class="existingWikiWord" href="/nlab/show/type">type</a>/<a class="existingWikiWord" href="/nlab/show/unit+type">unit type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/false">false</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/empty+set">empty set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/empty+type">empty type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/proposition">proposition</a>, <a class="existingWikiWord" href="/nlab/show/truth+value">truth value</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/subsingleton">subsingleton</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/subterminal+object">subterminal object</a>/<a class="existingWikiWord" href="/nlab/show/%28-1%29-truncated+object">(-1)-truncated object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/h-proposition">h-proposition</a>, <a class="existingWikiWord" href="/nlab/show/mere+proposition">mere proposition</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/logical+conjunction">logical conjunction</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cartesian+product">cartesian product</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/product">product</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/product+type">product type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/disjunction">disjunction</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a> (<a class="existingWikiWord" href="/nlab/show/support">support</a> of)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> (<a class="existingWikiWord" href="/nlab/show/%28-1%29-truncation">(-1)-truncation</a> of)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/sum+type">sum type</a> (<a class="existingWikiWord" href="/nlab/show/bracket+type">bracket type</a> of)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/implication">implication</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/function+set">function set</a> (into <a class="existingWikiWord" href="/nlab/show/subsingleton">subsingleton</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> (into <a class="existingWikiWord" href="/nlab/show/subterminal+object">subterminal object</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/function+type">function type</a> (into <a class="existingWikiWord" href="/nlab/show/h-proposition">h-proposition</a>)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/negation">negation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/function+set">function set</a> into <a class="existingWikiWord" href="/nlab/show/empty+set">empty set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> into <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/function+type">function type</a> into <a class="existingWikiWord" href="/nlab/show/empty+type">empty type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/universal+quantification">universal quantification</a></td><td style="text-align: left;">indexed <a class="existingWikiWord" href="/nlab/show/cartesian+product">cartesian product</a> (of family of <a class="existingWikiWord" href="/nlab/show/subsingletons">subsingletons</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a> (of family of <a class="existingWikiWord" href="/nlab/show/subterminal+objects">subterminal objects</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+product+type">dependent product type</a> (of family of <a class="existingWikiWord" href="/nlab/show/h-propositions">h-propositions</a>)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/existential+quantification">existential quantification</a></td><td style="text-align: left;">indexed <a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a> (<a class="existingWikiWord" href="/nlab/show/support">support</a> of)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a> (<a class="existingWikiWord" href="/nlab/show/%28-1%29-truncation">(-1)-truncation</a> of)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+sum+type">dependent sum type</a> (<a class="existingWikiWord" href="/nlab/show/bracket+type">bracket type</a> of)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/logical+equivalence">logical equivalence</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/bijection+set">bijection set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/object+of+isomorphisms">object of isomorphisms</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/equivalence+type">equivalence type</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/support+set">support set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/support+object">support object</a>/<a class="existingWikiWord" href="/nlab/show/%28-1%29-truncation">(-1)-truncation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/propositional+truncation">propositional truncation</a>/<a class="existingWikiWord" href="/nlab/show/bracket+type">bracket type</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/n-image">n-image</a> of <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> into <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>/<a class="existingWikiWord" href="/nlab/show/n-truncation">n-truncation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/n-truncation+modality">n-truncation modality</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/equality">equality</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/diagonal+function">diagonal function</a>/<a class="existingWikiWord" href="/nlab/show/diagonal+subset">diagonal subset</a>/<a class="existingWikiWord" href="/nlab/show/diagonal+relation">diagonal relation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/identity+type">identity type</a>/<a class="existingWikiWord" href="/nlab/show/path+type">path type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/completely+presented+set">completely presented set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/set">set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/discrete+object">discrete object</a>/<a class="existingWikiWord" href="/nlab/show/0-truncated+object">0-truncated object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/h-level+2">h-level 2</a>-<a class="existingWikiWord" href="/nlab/show/type">type</a>/<a class="existingWikiWord" href="/nlab/show/set">set</a>/<a class="existingWikiWord" href="/nlab/show/h-set">h-set</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/set">set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/set">set</a> with <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28infinity%2C1%29-category">internal 0-groupoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Bishop+set">Bishop set</a>/<a class="existingWikiWord" href="/nlab/show/setoid">setoid</a> with its <a class="existingWikiWord" href="/nlab/show/pseudo-equivalence+relation">pseudo-equivalence relation</a> an actual <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/equivalence+class">equivalence class</a>/<a class="existingWikiWord" href="/nlab/show/quotient+set">quotient set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quotient">quotient</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quotient+type">quotient type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/induction">induction</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/inductive+type">inductive type</a>, <a class="existingWikiWord" href="/nlab/show/W-type">W-type</a>, <a class="existingWikiWord" href="/nlab/show/M-type">M-type</a></td></tr> <tr><td style="text-align: left;">higher <a class="existingWikiWord" href="/nlab/show/induction">induction</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-colimit">higher colimit</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/higher+inductive+type">higher inductive type</a></td></tr> <tr><td style="text-align: left;">-</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/0-truncated">0-truncated</a> <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-colimit">higher colimit</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quotient+inductive+type">quotient inductive type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coinduction">coinduction</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/limit">limit</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coinductive+type">coinductive type</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/preset">preset</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type">type</a> without <a class="existingWikiWord" href="/nlab/show/identity+types">identity types</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/set">set</a> of <a class="existingWikiWord" href="/nlab/show/truth+values">truth values</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+of+propositions">type of propositions</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/domain+of+discourse">domain of discourse</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/universe">universe</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/object+classifier">object classifier</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+universe">type universe</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/modality">modality</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/closure+operator">closure operator</a>, (<a class="existingWikiWord" href="/nlab/show/idempotent+monad">idempotent</a>) <a class="existingWikiWord" href="/nlab/show/monad">monad</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/modal+type+theory">modal type theory</a>, <a class="existingWikiWord" href="/nlab/show/monad+%28in+computer+science%29">monad (in computer science)</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a></td><td style="text-align: left;"></td><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric</a>, <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed</a>) <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+type+theory">linear type theory</a>/<a class="existingWikiWord" href="/nlab/show/quantum+computation">quantum computation</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/proof+net">proof net</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/string+diagram">string diagram</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quantum+circuit">quantum circuit</a></td></tr> <tr><td style="text-align: left;">(absence of) <a class="existingWikiWord" href="/nlab/show/contraction+rule">contraction rule</a></td><td style="text-align: left;"></td><td style="text-align: left;">(absence of) <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/no-cloning+theorem">no-cloning theorem</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/synthetic+mathematics">synthetic mathematics</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/domain+specific+embedded+programming+language">domain specific embedded programming language</a></td></tr> </tbody></table> </div> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+levels">homotopy levels</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-type+theory">2-type theory</a>, <a class="existingWikiWord" href="/michaelshulman/show/2-categorical+logic">2-categorical logic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory+-+contents">homotopy type theory - contents</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/univalence">univalence</a>, <a class="existingWikiWord" href="/nlab/show/function+extensionality">function extensionality</a>, <a class="existingWikiWord" href="/nlab/show/internal+logic+of+an+%28%E2%88%9E%2C1%29-topos">internal logic of an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+type+theory">cohesive homotopy type theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/directed+homotopy+type+theory">directed homotopy type theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/HoTT+methods+for+homotopy+theorists">HoTT methods for homotopy theorists</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/semantics">semantics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a>, <a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/display+map">display map</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic+of+a+topos">internal logic of a topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Mitchell-Benabou+language">Mitchell-Benabou language</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kripke-Joyal+semantics">Kripke-Joyal semantics</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic+of+an+%28%E2%88%9E%2C1%29-topos">internal logic of an (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/type-theoretic+model+category">type-theoretic model category</a></li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/type+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="deduction_and_induction">Deduction and Induction</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/deductive+reasoning">deductive reasoning</a></strong>, <a class="existingWikiWord" href="/nlab/show/deduction">deduction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sequent">sequent</a></p> <p><a class="existingWikiWord" href="/nlab/show/hypothesis">hypothesis</a>/<a class="existingWikiWord" href="/nlab/show/context">context</a>/<a class="existingWikiWord" href="/nlab/show/antecedent">antecedent</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_77a3b8c239221bcb6b2f5d78b1dd70489c654f80_1"><semantics><mrow><mo>⊢</mo></mrow><annotation encoding="application/x-tex">\vdash</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/conclusion">conclusion</a>/<a class="existingWikiWord" href="/nlab/show/consequence">consequence</a>/<a class="existingWikiWord" href="/nlab/show/succedent">succedent</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/logical+framework">logical framework</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/deductive+system">deductive system</a>,</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+deduction">natural deduction</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequent+calculus">sequent calculus</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/inductive+reasoning">inductive reasoning</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/induction">induction</a>, <a class="existingWikiWord" href="/nlab/show/recursion">recursion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inductive+type">inductive type</a>, <a class="existingWikiWord" href="/nlab/show/higher+inductive+type">higher inductive type</a></p> </li> </ul></div></div> <h4 id="foundations">Foundations</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/foundations">foundations</a></strong></p> <h2 id="the_basis_of_it_all">The basis of it all</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/mathematical+logic">mathematical logic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/deduction+system">deduction system</a>, <a class="existingWikiWord" href="/nlab/show/natural+deduction">natural deduction</a>, <a class="existingWikiWord" href="/nlab/show/sequent+calculus">sequent calculus</a>, <a class="existingWikiWord" href="/nlab/show/lambda-calculus">lambda-calculus</a>, <a class="existingWikiWord" href="/nlab/show/judgment">judgment</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a>, <a class="existingWikiWord" href="/nlab/show/simple+type+theory">simple type theory</a>, <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/collection">collection</a>, <a class="existingWikiWord" href="/nlab/show/object">object</a>, <a class="existingWikiWord" href="/nlab/show/type">type</a>, <a class="existingWikiWord" href="/nlab/show/term">term</a>, <a class="existingWikiWord" href="/nlab/show/set">set</a>, <a class="existingWikiWord" href="/nlab/show/element">element</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equality">equality</a>, <a class="existingWikiWord" href="/nlab/show/judgmental+equality">judgmental equality</a>, <a class="existingWikiWord" href="/nlab/show/typal+equality">typal equality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universe">universe</a>, <a class="existingWikiWord" href="/nlab/show/size+issues">size issues</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher-order+logic">higher-order logic</a></p> </li> </ul> <h2 id="set_theory"> Set theory</h2> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/set+theory">set theory</a></strong></p> <ul> <li>fundamentals of set theory <ul> <li><a class="existingWikiWord" href="/nlab/show/propositional+logic">propositional logic</a></li> <li><a class="existingWikiWord" href="/nlab/show/first-order+logic">first-order logic</a></li> <li><a class="existingWikiWord" href="/nlab/show/typed+predicate+logic">typed predicate logic</a></li> <li><a class="existingWikiWord" href="/nlab/show/membership+relation">membership relation</a></li> <li><a class="existingWikiWord" href="/nlab/show/propositional+equality">propositional equality</a></li> <li><a class="existingWikiWord" href="/nlab/show/set">set</a>, <a class="existingWikiWord" href="/nlab/show/element">element</a>, <a class="existingWikiWord" href="/nlab/show/function">function</a>, <a class="existingWikiWord" href="/nlab/show/relation">relation</a></li> <li><a class="existingWikiWord" href="/nlab/show/universe">universe</a>, <a class="existingWikiWord" href="/nlab/show/small+set">small set</a>, <a class="existingWikiWord" href="/nlab/show/large+set">large set</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/material+set+theory">material set theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/membership+relation">membership relation</a>, <a class="existingWikiWord" href="/nlab/show/propositional+equality">propositional equality</a>, <a class="existingWikiWord" href="/nlab/show/axiom+of+extensionality">axiom of extensionality</a></li> <li><a class="existingWikiWord" href="/nlab/show/pairing+structure">pairing structure</a>, <a class="existingWikiWord" href="/nlab/show/axiom+of+pairing">axiom of pairing</a></li> <li><a class="existingWikiWord" href="/nlab/show/union+structure">union structure</a>, <a class="existingWikiWord" href="/nlab/show/axiom+of+union">axiom of union</a></li> <li><a class="existingWikiWord" href="/nlab/show/powerset+structure">powerset structure</a>, <a class="existingWikiWord" href="/nlab/show/axiom+of+power+sets">axiom of power sets</a></li> <li><a class="existingWikiWord" href="/nlab/show/natural+numbers+structure">natural numbers structure</a>, <a class="existingWikiWord" href="/nlab/show/axiom+of+infinity">axiom of infinity</a></li> </ul> </li> <li>presentations of set theory <ul> <li><a class="existingWikiWord" href="/nlab/show/first-order+set+theory">first-order set theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/unsorted+set+theory">unsorted set theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/simply+sorted+set+theory">simply sorted set theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/one-sorted+set+theory">one-sorted set theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/two-sorted+set+theory">two-sorted set theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/three-sorted+set+theory">three-sorted set theory</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/dependently+sorted+set+theory">dependently sorted set theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/structurally+presented+set+theory">structurally presented set theory</a></li> </ul> </li> <li>structuralism in set theory <ul> <li><a class="existingWikiWord" href="/nlab/show/material+set+theory">material set theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/ZFC">ZFC</a></li> <li><a class="existingWikiWord" href="/nlab/show/ZFA">ZFA</a></li> <li><a class="existingWikiWord" href="/nlab/show/Mostowski+set+theory">Mostowski set theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/New+Foundations">New Foundations</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/structural+set+theory">structural set theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/categorical+set+theory">categorical set theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/ETCS">ETCS</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/fully+formal+ETCS">fully formal ETCS</a></li> <li><a class="existingWikiWord" href="/nlab/show/ETCS+with+elements">ETCS with elements</a></li> <li><a class="existingWikiWord" href="/nlab/show/Trimble+on+ETCS+I">Trimble on ETCS I</a></li> <li><a class="existingWikiWord" href="/nlab/show/Trimble+on+ETCS+II">Trimble on ETCS II</a></li> <li><a class="existingWikiWord" href="/nlab/show/Trimble+on+ETCS+III">Trimble on ETCS III</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/structural+ZFC">structural ZFC</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/allegorical+set+theory">allegorical set theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/SEAR">SEAR</a></li> </ul> </li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/class-set+theory">class-set theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/class">class</a>, <a class="existingWikiWord" href="/nlab/show/proper+class">proper class</a></li> <li><a class="existingWikiWord" href="/nlab/show/universal+class">universal class</a>, <a class="existingWikiWord" href="/nlab/show/universe">universe</a></li> <li><a class="existingWikiWord" href="/nlab/show/category+of+classes">category of classes</a></li> <li><a class="existingWikiWord" href="/nlab/show/category+with+class+structure">category with class structure</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/constructive+set+theory">constructive set theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/algebraic+set+theory">algebraic set theory</a></li> </ul> </div> <h2 id="foundational_axioms">Foundational axioms</h2> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/foundational+axiom">foundational</a> <a class="existingWikiWord" href="/nlab/show/axioms">axioms</a></strong></p> <ul> <li> <p>basic constructions:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+cartesian+products">axiom of cartesian products</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+disjoint+unions">axiom of disjoint unions</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+the+empty+set">axiom of the empty set</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+fullness">axiom of fullness</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+function+sets">axiom of function sets</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+power+sets">axiom of power sets</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+quotient+sets">axiom of quotient sets</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/material+set+theory">material axioms</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+extensionality">axiom of extensionality</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+foundation">axiom of foundation</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+anti-foundation">axiom of anti-foundation</a></li> <li><a class="existingWikiWord" href="/nlab/show/Mostowski%27s+axiom">Mostowski's axiom</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+pairing">axiom of pairing</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+transitive+closure">axiom of transitive closure</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+union">axiom of union</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/structural+set+theory">structural axioms</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+materialization">axiom of materialization</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/type+theory">type theoretic axioms</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/axioms+of+set+truncation">axioms of set truncation</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/uniqueness+of+identity+proofs">uniqueness of identity proofs</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+K">axiom K</a></li> <li><a class="existingWikiWord" href="/nlab/show/boundary+separation">boundary separation</a></li> <li><a class="existingWikiWord" href="/nlab/show/equality+reflection">equality reflection</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+circle+type+localization">axiom of circle type localization</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theoretic axioms</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/univalence+axiom">univalence axiom</a></li> <li><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+principle">Whitehead's principle</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/axioms+of+choice">axioms of choice</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+countable+choice">axiom of countable choice</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+dependent+choice">axiom of dependent choice</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+excluded+middle">axiom of excluded middle</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+existence">axiom of existence</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+multiple+choice">axiom of multiple choice</a></li> <li><a class="existingWikiWord" href="/nlab/show/Markov%27s+axiom">Markov's axiom</a></li> <li><a class="existingWikiWord" href="/nlab/show/presentation+axiom">presentation axiom</a></li> <li><a class="existingWikiWord" href="/nlab/show/small+cardinality+selection+axiom">small cardinality selection axiom</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+small+violations+of+choice">axiom of small violations of choice</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+weakly+initial+sets+of+covers">axiom of weakly initial sets of covers</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/large+cardinal+axioms">large cardinal axioms</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+infinity">axiom of infinity</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+universes">axiom of universes</a></li> <li><a class="existingWikiWord" href="/nlab/show/regular+extension+axiom">regular extension axiom</a></li> <li><a class="existingWikiWord" href="/nlab/show/inaccessible+cardinal">inaccessible cardinal</a></li> <li><a class="existingWikiWord" href="/nlab/show/measurable+cardinal">measurable cardinal</a></li> <li><a class="existingWikiWord" href="/nlab/show/elementary+embedding">elementary embedding</a></li> <li><a class="existingWikiWord" href="/nlab/show/supercompact+cardinal">supercompact cardinal</a></li> <li><a class="existingWikiWord" href="/nlab/show/Vop%C4%9Bnka%27s+principle">Vopěnka's principle</a></li> </ul> </li> <li> <p>strong axioms</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+separation">axiom of separation</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+replacement">axiom of replacement</a></li> </ul> </li> <li> <p>further</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/reflection+principle">reflection principle</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/axiom+of+tight+apartness">axiom of tight apartness</a></p> </li> </ul> </div> <h2 id="removing_axioms">Removing axioms</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a></li> <li><a class="existingWikiWord" href="/nlab/show/predicative+mathematics">predicative mathematics</a></li> </ul> <div> <p> <a href="/nlab/edit/foundations+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#with_typal_computation_and_uniqueness_rules'>With typal computation and uniqueness rules</a></li> <li><a href='#generalized_induction_principle'>Generalized induction principle</a></li> <li><a href='#extensionality_principle_of_the_natural_numbers'>Extensionality principle of the natural numbers</a></li> <li><a href='#large_recursion_principle'>Large recursion principle</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#relation_to_the_type_of_finite_types'>Relation to the type of finite types</a></li> <li><a href='#CategoricalSemantics'>Categorical semantics</a></li> <ul> <li><a href='#Recursion'>Recursion</a></li> <li><a href='#Induction'>Induction</a></li> </ul> </ul> <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/type+theory">type theory</a>: the the <em>natural numbers type</em> is the <a class="existingWikiWord" href="/nlab/show/type">type</a> of <a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</a>.</p> <h2 id="definition">Definition</h2> <div class="num_defn" id="InferenceRules"> <h6 id="definition_2">Definition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/type+of+natural+numbers">type of natural numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">\mathbb{N}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/inductive+type">inductive type</a> defined by the following <a class="existingWikiWord" href="/nlab/show/inference+rules">inference rules</a>.</p> <ol> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/type+formation+rule">type formation rule</a></strong>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow></mrow><mrow><mpadded width="0" lspace="-50%width"><mphantom><mrow><msup><mo stretchy="false">|</mo> <mo stretchy="false">|</mo></msup></mrow></mphantom></mpadded><mi>ℕ</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>Type</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex"> \frac{}{ \mathclap{\phantom{\vert^{\vert}}} \mathbb{N} \,\colon\, Type } </annotation></semantics></math></div></li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/term+introduction+rules">term introduction rules</a></strong>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow></mrow><mrow><mpadded width="0" lspace="-50%width"><mphantom><mrow><msup><mo stretchy="false">|</mo> <mo stretchy="false">|</mo></msup></mrow></mphantom></mpadded><mn>0</mn><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>ℕ</mi></mrow></mfrac><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mfrac><mrow><mi>n</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>ℕ</mi><mpadded width="0" lspace="-50%width"><mphantom><mrow><msub><mo stretchy="false">|</mo> <mo stretchy="false">|</mo></msub></mrow></mphantom></mpadded></mrow><mrow><mpadded width="0" lspace="-50%width"><mphantom><mrow><msup><mo stretchy="false">|</mo> <mo stretchy="false">|</mo></msup></mrow></mphantom></mpadded><mi>succ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>ℕ</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex"> \frac{}{ \mathclap{\phantom{\vert^{\vert}}} 0 \,\colon\, \mathbb{N} } \;\;\;\;\;\;\;\; \frac{ n \,\colon\, \mathbb{N} \mathclap{\phantom{\vert_{\vert}}} }{ \mathclap{\phantom{\vert^{\vert}}} succ(n) \,\colon\, \mathbb{N} } </annotation></semantics></math></div></li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/term+elimination+rule">term elimination rule</a></strong>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>n</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>ℕ</mi><mspace width="thickmathspace"></mspace><mo>⊢</mo><mspace width="thickmathspace"></mspace><mi>P</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>Type</mi><mspace width="thickmathspace"></mspace><mo>;</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊢</mo><mspace width="thickmathspace"></mspace><msub><mn>0</mn> <mi>P</mi></msub><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>P</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>;</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>n</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>ℕ</mi><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mi>p</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>⊢</mo><mspace width="thickmathspace"></mspace><msub><mi>succ</mi> <mi>P</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>p</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>P</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>succ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mpadded width="0" lspace="-50%width"><mphantom><mrow><msub><mo stretchy="false">|</mo> <mo stretchy="false">|</mo></msub></mrow></mphantom></mpadded></mrow><mrow><mpadded width="0" lspace="-50%width"><mphantom><mrow><msup><mo stretchy="false">|</mo> <mo stretchy="false">|</mo></msup></mrow></mphantom></mpadded><mi>n</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>ℕ</mi><mspace width="thickmathspace"></mspace><mo>⊢</mo><mspace width="thickmathspace"></mspace><msub><mi>ind</mi> <mrow><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><msub><mn>0</mn> <mi>P</mi></msub><mo>,</mo><msub><mi>succ</mi> <mi>P</mi></msub><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>P</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex"> \frac{ n \,\colon\, \mathbb{N} \;\vdash\; P(n) \,\colon\, Type \;; \;\;\;\; \vdash\; 0_P \,\colon\, P(0) \;; \;\;\;\; n \,\colon\, \mathbb{N} \,, \; p \,\colon\, P(x) \;\vdash\; succ_P(n,\,p) \,\colon\, P\big(succ(n)\big) \mathclap{\phantom{\vert_{\vert}}} }{ \mathclap{\phantom{\vert^{\vert}}} n \,\colon\, \mathbb{N} \;\vdash\; ind_{(P, 0_P, succ_P)}(n) \,\colon\, P(n) } </annotation></semantics></math></div></li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/computation+rules">computation rules</a></strong>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>n</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>ℕ</mi><mspace width="thickmathspace"></mspace><mo>⊢</mo><mspace width="thickmathspace"></mspace><mi>P</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>Type</mi><mspace width="thickmathspace"></mspace><mo>;</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊢</mo><mspace width="thickmathspace"></mspace><msub><mn>0</mn> <mi>P</mi></msub><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>P</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>;</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>n</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>ℕ</mi><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mi>p</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>⊢</mo><mspace width="thickmathspace"></mspace><msub><mi>succ</mi> <mi>P</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>P</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>succ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mpadded width="0" lspace="-50%width"><mphantom><mrow><msub><mo stretchy="false">|</mo> <mo stretchy="false">|</mo></msub></mrow></mphantom></mpadded></mrow><mrow><mpadded width="0" lspace="-50%width"><mphantom><mrow><msup><mo stretchy="false">|</mo> <mo stretchy="false">|</mo></msup></mrow></mphantom></mpadded><msub><mi>ind</mi> <mrow><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><msub><mn>0</mn> <mi>P</mi></msub><mo>,</mo><msub><mi>succ</mi> <mi>P</mi></msub><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>=</mo><mspace width="thinmathspace"></mspace><msub><mn>0</mn> <mi>P</mi></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex"> \frac{ n \,\colon\, \mathbb{N} \;\vdash\; P(n) \,\colon\, Type \;; \;\;\;\; \vdash\; 0_P \,\colon\, P(0) \;; \;\;\;\; n \,\colon\, \mathbb{N} \,, \; p \,\colon\, P(x) \;\vdash\; succ_P(n,p) \,\colon\, P\big(succ(n)\big) \mathclap{\phantom{\vert_{\vert}}} }{ \mathclap{\phantom{\vert^{\vert}}} ind_{(P, 0_P, succ_P)}(0) \,=\, 0_P } </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>n</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>ℕ</mi><mspace width="thickmathspace"></mspace><mo>⊢</mo><mspace width="thickmathspace"></mspace><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>Type</mi><mspace width="thickmathspace"></mspace><mo>;</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊢</mo><mspace width="thickmathspace"></mspace><msub><mn>0</mn> <mi>P</mi></msub><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>P</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>;</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>n</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>ℕ</mi><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mi>p</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>⊢</mo><mspace width="thickmathspace"></mspace><msub><mi>succ</mi> <mi>P</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>P</mi><mo stretchy="false">(</mo><mi>s</mi><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>;</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊢</mo><mspace width="thickmathspace"></mspace><mi>n</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>ℕ</mi><mpadded width="0" lspace="-50%width"><mphantom><mrow><msub><mo stretchy="false">|</mo> <mo stretchy="false">|</mo></msub></mrow></mphantom></mpadded></mrow><mrow><mpadded width="0" lspace="-50%width"><mphantom><mrow><msup><mo stretchy="false">|</mo> <mo stretchy="false">|</mo></msup></mrow></mphantom></mpadded><msub><mi>ind</mi> <mrow><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><msub><mn>0</mn> <mi>P</mi></msub><mo>,</mo><msub><mi>succ</mi> <mi>P</mi></msub><mo stretchy="false">)</mo></mrow></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>succ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>=</mo><mspace width="thinmathspace"></mspace><msub><mi>succ</mi> <mi>P</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>n</mi><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>ind</mi> <mrow><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><msub><mn>0</mn> <mi>P</mi></msub><mo>,</mo><msub><mi>succ</mi> <mi>P</mi></msub><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex"> \frac{ n \,\colon\, \mathbb{N} \;\vdash\; P(x) \,\colon\, Type \;; \;\;\;\; \vdash\; 0_P \,\colon\, P(0) \;; \;\;\;\; n \,\colon\, \mathbb{N} \,, \; p \,\colon\, P(x) \;\vdash\; succ_P(x,p) \,\colon\, P(s x) \;; \;\;\;\; \vdash\; n \,\colon\, \mathbb{N} \mathclap{\phantom{\vert_{\vert}}} }{ \mathclap{\phantom{\vert^{\vert}}} ind_{(P, 0_P, succ_P)}\big(succ(n)\big) \,=\, succ_P\big(n,\, ind_{(P, 0_P, succ_P)}(n) \big) } </annotation></semantics></math></div></li> </ol> <p>(In the last line, “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>=</mo></mrow><annotation encoding="application/x-tex">=</annotation></semantics></math>” denotes <a class="existingWikiWord" href="/nlab/show/judgemental+equality">judgemental equality</a>.)</p> </div> <p>That this is the right definition (and a special case of the general principle of <a class="existingWikiWord" href="/nlab/show/inductive+types">inductive types</a>) was clearly understood around <a href="#Martin-Löf84">Martin-Löf (1984)</a>, <a href="/nlab/files/MartinLofIntuitionisticTypeTheory.pdf#page=44">pp. 38</a>; <a href="#CoquandPaulin90">Coquand & Paulin (1990, p. 52-53)</a>; <a href="#Paulin-Mohring93">Paulin-Mohring (1993, §1.3)</a>; <a href="#Dybjer94">Dybjer (1994, §3)</a>. For review see also, e.g., <a href="#Pfenning">Pfenning (2009, §2)</a>; <a href="#UFP13">UFP (2013, §1.9)</a>; <a href="#Söhnen18">Söhnen (2018, §2.4.5)</a>.</p> <p>In <a class="existingWikiWord" href="/nlab/show/Coq">Coq</a>-<a class="existingWikiWord" href="/nlab/show/syntax">syntax</a> the <a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</a> are the <a class="existingWikiWord" href="/nlab/show/inductive+type">inductive type</a> defined [cf. <a href="#Paulin-Mohring14">Paulin-Mohring (2014, p. 6)</a>] by:</p> <pre><code>Inductive nat : Type := | zero : nat | succ : nat -> nat.</code></pre> <p>In the <a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a> (via the <a class="existingWikiWord" href="/nlab/show/categorical+model+of+dependent+types">categorical model of dependent types</a>, see <a href="#CategoricalSemantics">below</a>) this is interpreted as the <a class="existingWikiWord" href="/nlab/show/initial+algebra+for+an+endofunctor">initial algebra</a> for the <a class="existingWikiWord" href="/nlab/show/endofunctor">endofunctor</a> <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> that sends an object to its <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> with the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a></p> <div class="maruku-equation" id="eq:TheEndofunctor"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo>*</mo><mo>⊔</mo><mi>X</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> F(X) \;\coloneqq\; * \sqcup X \,, </annotation></semantics></math></div> <p>or in different equivalent notation, which is very suggestive here:</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 stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mn>1</mn><mo>+</mo><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> F(X) \;=\; 1 + X \,. </annotation></semantics></math></div> <p>That <a class="existingWikiWord" href="/nlab/show/initial+algebra+for+an+endofunctor">initial algebra</a> is (as also explained <a href="#initial+algebra+of+an+endofunctor#NaturalNumbers">there</a>) precisely a <a class="existingWikiWord" href="/nlab/show/natural+number+object">natural number object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">\mathbb{N}</annotation></semantics></math>. The two components of the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>ℕ</mi><mo stretchy="false">)</mo><mo>→</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">F(\mathbb{N}) \to \mathbb{N}</annotation></semantics></math> that defines the algebra structure are the 0-<a class="existingWikiWord" href="/nlab/show/generalized+element">element</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mo>*</mo><mo>→</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">0 \,\colon\, * \to \mathbb{N}</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/successor">successor</a> <a class="existingWikiWord" href="/nlab/show/endomorphism">endomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>succ</mi><mo lspace="verythinmathspace">:</mo><mi>ℕ</mi><mo>→</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">succ \colon \mathbb{N} \to \mathbb{N}</annotation></semantics></math></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><mn>0</mn><mo>,</mo><mspace width="thinmathspace"></mspace><mi>succ</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo>*</mo><mo>⊔</mo><mi>ℕ</mi><mo>⟶</mo><mi>ℕ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (0 ,\, succ) \;\colon\; * \sqcup \mathbb{N} \longrightarrow \mathbb{N} \,. </annotation></semantics></math></div> <h3 id="with_typal_computation_and_uniqueness_rules">With typal computation and uniqueness rules</h3> <p>Assuming that <a class="existingWikiWord" href="/nlab/show/identification+types">identification types</a>, <a class="existingWikiWord" href="/nlab/show/function+types">function types</a> and <a class="existingWikiWord" href="/nlab/show/dependent+sequence+types">dependent sequence types</a> exist in the type theory, the natural numbers type is the <a class="existingWikiWord" href="/nlab/show/inductive+type">inductive type</a> generated by an element and a <a class="existingWikiWord" href="/nlab/show/function">function</a>:</p> <p>Formation rules for the natural numbers type:</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><mspace width="thickmathspace"></mspace><mi mathvariant="normal">ctx</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><mi>ℕ</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathbb{N} \; \mathrm{type}}</annotation></semantics></math></div> <p>Introduction rules for the natural numbers type:</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><mspace width="thickmathspace"></mspace><mi mathvariant="normal">ctx</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><mn>0</mn><mo>:</mo><mi>ℕ</mi></mrow></mfrac><mspace width="2em"></mspace><mfrac><mrow><mi>Γ</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">ctx</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><mi>s</mi><mo>:</mo><mi>ℕ</mi><mo>→</mo><mi>ℕ</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash 0:\mathbb{N}} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash s:\mathbb{N} \to \mathbb{N}}</annotation></semantics></math></div> <p>Elimination rules for the natural numbers type:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>ℕ</mi><mo>⊢</mo><mi>C</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><msub><mi>c</mi> <mn>0</mn></msub><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>n</mi><mo>:</mo><mi>ℕ</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><msubsup><mi mathvariant="normal">ind</mi> <mi>ℕ</mi> <mi>C</mi></msubsup><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma, x:\mathbb{N} \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_0:C(0) \quad \Gamma \vdash c_s:\prod_{x:\mathbb{N}} C(x) \to C(s(x)) \quad \Gamma \vdash n:\mathbb{N}}{\Gamma \vdash \mathrm{ind}_\mathbb{N}^C(c_0, c_s, n):C(n)}</annotation></semantics></math></div> <p>Computation rules for the natural numbers type:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>ℕ</mi><mo>⊢</mo><mi>C</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><msub><mi>c</mi> <mn>0</mn></msub><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>Γ</mi><mo>⊢</mo><msubsup><mi>β</mi> <mi>ℕ</mi> <mn>0</mn></msubsup><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo stretchy="false">)</mo><mo>:</mo><msub><mi mathvariant="normal">Id</mi> <mrow><mi>C</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msub><msubsup><mi mathvariant="normal">ind</mi> <mi>ℕ</mi> <mi>C</mi></msubsup><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo><msub><mi>c</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma, x:\mathbb{N} \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_0:C(0) \quad \Gamma \vdash c_s:\prod_{x:\mathbb{N}} C(x) \to C(s(x))}{\Gamma \vdash \beta_\mathbb{N}^0(c_0, c_s):\mathrm{Id}_{C(0)}\mathrm{ind}_\mathbb{N}^C(c_0, c_s, 0), c_0)}</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>ℕ</mi><mo>⊢</mo><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>n</mi><mo>:</mo><mi>ℕ</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><msubsup><mi>β</mi> <mi>ℕ</mi> <mi>s</mi></msubsup><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo>:</mo><msub><mi mathvariant="normal">Id</mi> <mrow><mi>C</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msubsup><mi mathvariant="normal">ind</mi> <mi>ℕ</mi> <mi>C</mi></msubsup><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><mi>s</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msubsup><mi mathvariant="normal">ind</mi> <mi>ℕ</mi> <mi>C</mi></msubsup><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma, x:\mathbb{N} \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_0:C(0) \quad \Gamma \vdash c_s:\prod_{x:\mathbb{N}} C(x) \to C(s(x)) \quad \Gamma \vdash n:\mathbb{N}}{\Gamma \vdash \beta_\mathbb{N}^s(c_0, c_s, n):\mathrm{Id}_{C(s(n))}(\mathrm{ind}_\mathbb{N}^C(c_0, c_s, s(n)), c_s(n)(\mathrm{ind}_\mathbb{N}^C(c_0, c_s, n)))}</annotation></semantics></math></div> <p>Uniqueness rules for the natural numbers type:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>ℕ</mi><mo>⊢</mo><mi>C</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>c</mi><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>n</mi><mo>:</mo><mi>ℕ</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><msub><mi>η</mi> <mi>ℕ</mi></msub><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo>:</mo><msub><mi mathvariant="normal">Id</mi> <mrow><mi>C</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msubsup><mi mathvariant="normal">ind</mi> <mi>ℕ</mi> <mi>C</mi></msubsup><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi>ℕ</mi><mo>.</mo><mi>c</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo>,</mo><mi>c</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma, x:\mathbb{N} \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c:\prod_{x:\mathbb{N}} C(x) \quad \Gamma \vdash n:\mathbb{N}}{\Gamma \vdash \eta_\mathbb{N}(c, n):\mathrm{Id}_{C(n)}(\mathrm{ind}_\mathbb{N}^C(c(0), \lambda x:\mathbb{N}.c(s(x)), n), c(n))}</annotation></semantics></math></div> <p>The elimination, computation, and uniqueness rules for the natural numbers type state that the natural numbers type satisfy the <strong>dependent universal property of the natural numbers</strong>. If the dependent type theory also has <a class="existingWikiWord" href="/nlab/show/dependent+sum+types">dependent sum types</a> and <a class="existingWikiWord" href="/nlab/show/product+types">product types</a>, allowing one to define the <a class="existingWikiWord" href="/nlab/show/uniqueness+quantifier">uniqueness quantifier</a>, the dependent universal property of the natural numbers could be simplified to the following rule:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>ℕ</mi><mo>⊢</mo><mi>C</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><msub><mi>c</mi> <mn>0</mn></msub><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>Γ</mi><mo>⊢</mo><msubsup><mi mathvariant="normal">up</mi> <mi>ℕ</mi> <mi>C</mi></msubsup><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo stretchy="false">)</mo><mo>:</mo><mo>∃</mo><mo>!</mo><mi>c</mi><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>.</mo><msub><mi mathvariant="normal">Id</mi> <mrow><mi>C</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo><msub><mi>c</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo>×</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><msub><mi mathvariant="normal">Id</mi> <mrow><mi>C</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma, x:\mathbb{N} \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_0:C(0) \quad \Gamma \vdash c_s:\prod_{x:\mathbb{N}} C(x) \to C(s(x))}{\Gamma \vdash \mathrm{up}_\mathbb{N}^C(c_0, c_s):\exists!c:\prod_{x:\mathbb{N}} C(x).\mathrm{Id}_{C(0)}(c(0), c_0) \times \prod_{x:\mathbb{N}} \mathrm{Id}_{C(s(x))}(c(s(x)), c_s(c(x)))}</annotation></semantics></math></div> <p>The dependent universal property of the natural numbers is used to characterize the <a class="existingWikiWord" href="/nlab/show/dependent+product+type">dependent product type</a> of an type family <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(x)</annotation></semantics></math> dependent on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">x:\mathbb{N}</annotation></semantics></math>, and states that the <a class="existingWikiWord" href="/nlab/show/fibers">fibers</a> of 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>c</mi><mo>.</mo><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo><mi>λ</mi><mi>x</mi><mo>.</mo><mi>c</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>→</mo><mrow><mo>(</mo><mi>C</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>×</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\lambda c.(c(0), \lambda x.c(s(x))):\prod_{x:\mathbb{N}} C(x) \to \left(C(0) \times \prod_{x:\mathbb{N}} C(x) \to C(s(x))\right)</annotation></semantics></math></div> <p>are <a class="existingWikiWord" href="/nlab/show/contractible+types">contractible types</a>. This is equivalent to saying that the above function 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><mi mathvariant="normal">isEquiv</mi><mo stretchy="false">(</mo><mi>λ</mi><mi>c</mi><mo>.</mo><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo><mi>λ</mi><mi>x</mi><mo>.</mo><mi>c</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{isEquiv}(\lambda c.(c(0), \lambda x.c(s(x))))</annotation></semantics></math></div> <p>The non-dependent universal property similarly says that given a 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> 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>c</mi><mo>.</mo><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo><mi>λ</mi><mi>x</mi><mo>.</mo><mi>c</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</mo><mi>ℕ</mi><mo>→</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><mrow><mo>(</mo><mi>C</mi><mo>×</mo><mo stretchy="false">(</mo><mi>ℕ</mi><mo>→</mo><mi>C</mi><mo>→</mo><mi>C</mi><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\lambda c.(c(0), \lambda x.c(s(x))):(\mathbb{N} \to C) \to \left(C \times (\mathbb{N} \to C \to C)\right)</annotation></semantics></math></div> <p>is an equivalence of 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">isEquiv</mi><mo stretchy="false">(</mo><mi>λ</mi><mi>c</mi><mo>.</mo><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo><mi>λ</mi><mi>x</mi><mo>.</mo><mi>c</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{isEquiv}(\lambda c.(c(0), \lambda x.c(s(x))))</annotation></semantics></math></div> <h3 id="generalized_induction_principle">Generalized induction principle</h3> <p>There is also a generalized induction principle (cf. the talk slides in <a href="#LumsdaineShulman17">LumsdaineShulman17</a>), which uses a 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 a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">f:C \to \mathbb{N}</annotation></semantics></math> instead of 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>ℕ</mi><mo>⊢</mo><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:\mathbb{N} \vdash P(x)</annotation></semantics></math>, and one uses the <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> <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>z</mi><mo>:</mo><mi>C</mi></mrow></msub><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>n</mi></mrow><annotation encoding="application/x-tex">\sum_{z:C} f(z) =_\mathbb{N} n</annotation></semantics></math> to express the generalized induction principle.</p> <p>Then the induction principle states that given a 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 a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">f:C \to \mathbb{N}</annotation></semantics></math> along with</p> <ul> <li>dependent pair</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>0</mn></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>z</mi><mo>:</mo><mi>C</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mn>0</mn></mrow><annotation encoding="application/x-tex">c_0:\sum_{z:C} f(z) =_\mathbb{N} 0</annotation></semantics></math></div> <ul> <li>dependent function</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mi>s</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>n</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>z</mi><mo>:</mo><mi>C</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>n</mi><mo>)</mo></mrow><mo>→</mo><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>z</mi><mo>:</mo><mi>C</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>s</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">c_s:\prod_{n:\mathbb{N}} \left(\sum_{z:C} f(z) =_\mathbb{N} n\right) \to \left(\sum_{z:C} f(z) =_\mathbb{N} s(n)\right)</annotation></semantics></math></div> <ul> <li>and natural number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>:</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n:\mathbb{N}</annotation></semantics></math></li> </ul> <p>one could construct the dependent pair</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">ind</mi> <mi>ℕ</mi> <mi>C</mi></msubsup><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>z</mi><mo>:</mo><mi>C</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>n</mi></mrow><annotation encoding="application/x-tex">\mathrm{ind}_\mathbb{N}^C(f, c_0, c_s, n):\sum_{z:C} f(z) =_\mathbb{N} n</annotation></semantics></math></div> <p>such that</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">ind</mi> <mi>ℕ</mi> <mi>C</mi></msubsup><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>≡</mo><msub><mi>c</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{ind}_\mathbb{N}^C(f, c_0, c_s, 0) \equiv c_0</annotation></semantics></math></div> <p>and for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>:</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n:\mathbb{N}</annotation></semantics></math></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">ind</mi> <mi>ℕ</mi> <mi>C</mi></msubsup><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><mi>s</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><msub><mi>c</mi> <mi>s</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi mathvariant="normal">ind</mi><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{ind}_\mathbb{N}^C(f, c_0, c_s, s(n)) \equiv c_s(n, \mathrm{ind}(f, c_0, c_s, n))</annotation></semantics></math></div> <p>However, by the rules of dependent pair types, one could instead postulate separate elements and identifications instead of an element of a <a class="existingWikiWord" href="/nlab/show/fiber+type">fiber type</a> throughout the generalized principle.</p> <p>Instead of the dependent pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>0</mn></msub><mo>:</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>z</mi><mo>:</mo><mi>C</mi></mrow></msub><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mn>0</mn></mrow><annotation encoding="application/x-tex">c_0:\sum_{z:C} f(z) =_\mathbb{N} 0</annotation></semantics></math> we have the element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>0</mn></msub><mo>:</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c_0:C</annotation></semantics></math> and identificaiton <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>0</mn></msub><mo>:</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mn>0</mn></mrow><annotation encoding="application/x-tex">p_0:f(c_0) =_\mathbb{N} 0</annotation></semantics></math>, where the original element is given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(c_0, p_0)</annotation></semantics></math>. In addition, given the dependent type</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mi>s</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>n</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>z</mi><mo>:</mo><mi>C</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>n</mi><mo>)</mo></mrow><mo>→</mo><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>z</mi><mo>:</mo><mi>C</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>s</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">c_s:\prod_{n:\mathbb{N}} \left(\sum_{z:C} f(z) =_\mathbb{N} n\right) \to \left(\sum_{z:C} f(z) =_\mathbb{N} s(n)\right)</annotation></semantics></math></div> <p>by currying this is equivalent to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mi>s</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>n</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>z</mi><mo>:</mo><mi>C</mi></mrow></munder><mrow><mo>(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>n</mi><mo>)</mo></mrow><mo>→</mo><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>z</mi><mo>:</mo><mi>C</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>s</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">c_s:\prod_{n:\mathbb{N}} \prod_{z:C} \left(f(z) =_\mathbb{N} n\right) \to \left(\sum_{z:C} f(z) =_\mathbb{N} s(n)\right)</annotation></semantics></math></div> <p>and by the <a class="existingWikiWord" href="/nlab/show/type+theoretic+axiom+of+choice">type theoretic axiom of choice</a> this is equivalent to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mi>s</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>n</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>y</mi><mo>:</mo><mi>C</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>g</mi><mo>:</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>p</mi><mo>:</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>n</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>s</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c_s:\prod_{n:\mathbb{N}} \prod_{y:C} \sum_{g:(f(y) =_\mathbb{N} n) \to C} \prod_{p:f(y) =_\mathbb{N} n} f(g(p)) =_\mathbb{N} s(n)</annotation></semantics></math></div> <p>By the rules of dependent pair types, the family of dependent pair types could be split up into</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mi>s</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>n</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>y</mi><mo>:</mo><mi>C</mi></mrow></munder><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c_s:\prod_{n:\mathbb{N}} \prod_{y:C} (f(y) =_\mathbb{N} n) \to C</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>s</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>n</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>y</mi><mo>:</mo><mi>C</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>p</mi><mo>:</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>n</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mi>s</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>s</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p_s:\prod_{n:\mathbb{N}} \prod_{y:C} \prod_{p:f(y) =_\mathbb{N} n} f(c_s(n, y, p)) =_\mathbb{N} s(n)</annotation></semantics></math></div> <p>where the original dependent funciton is given by</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>n</mi><mo>:</mo><mi>ℕ</mi><mo>.</mo><mi>λ</mi><mi>y</mi><mo>:</mo><mi>C</mi><mo>.</mo><mo stretchy="false">(</mo><msub><mi>c</mi> <mi>s</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>p</mi> <mi>s</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lambda n:\mathbb{N}.\lambda y:C.(c_s(n, y), p_s(n, y))</annotation></semantics></math></div> <p>Then the induction principle of the natural numbers states that given a 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 a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">f:C \to \mathbb{N}</annotation></semantics></math>, along with</p> <ul> <li> <p>an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>0</mn></msub><mo>:</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c_0:C</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><msub><mi>p</mi> <mn>0</mn></msub><mo>:</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mn>0</mn></mrow><annotation encoding="application/x-tex">p_0:f(c_0) =_\mathbb{N} 0</annotation></semantics></math></p> </li> <li> <p>dependent functions</p> </li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mi>s</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>n</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>y</mi><mo>:</mo><mi>C</mi></mrow></munder><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c_s:\prod_{n:\mathbb{N}} \prod_{y:C} (f(y) =_\mathbb{N} n) \to C</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>s</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>n</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>y</mi><mo>:</mo><mi>C</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>p</mi><mo>:</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>n</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mi>s</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>s</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p_s:\prod_{n:\mathbb{N}} \prod_{y:C} \prod_{p:f(y) =_\mathbb{N} n} f(c_s(n, y, p)) =_\mathbb{N} s(n)</annotation></semantics></math></div> <p>we have a function</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">ind</mi> <mi>ℕ</mi> <mi>C</mi></msubsup><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><msub><mi>p</mi> <mi>s</mi></msub><mo stretchy="false">)</mo><mo>:</mo><mi>ℕ</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">\mathrm{ind}_\mathbb{N}^{C}(f, c_0, p_0, c_s, p_s):\mathbb{N} \to C</annotation></semantics></math></div> <p>and a homotopy</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">ind</mi> <mi>ℕ</mi> <mrow><mi>C</mi><mo>,</mo><mi mathvariant="normal">sec</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><msub><mi>p</mi> <mi>s</mi></msub><mo stretchy="false">)</mo><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>n</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi mathvariant="normal">ind</mi> <mi>ℕ</mi> <mi>C</mi></msubsup><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><msub><mi>p</mi> <mi>s</mi></msub><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>n</mi></mrow><annotation encoding="application/x-tex">\mathrm{ind}_\mathbb{N}^{C, \mathrm{sec}}(f, c_0, p_0, c_s, p_s):\prod_{n:\mathbb{N}} f(\mathrm{ind}_\mathbb{N}^{C}(f, c_0, p_0, c_s, p_s, n)) =_\mathbb{N} n</annotation></semantics></math></div> <p>indicating that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi mathvariant="normal">ind</mi> <mi>ℕ</mi> <mi>C</mi></msubsup><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><msub><mi>p</mi> <mi>s</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{ind}_\mathbb{N}^{C}(f, c_0, p_0, c_s, p_s)</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>, such that</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">ind</mi> <mi>ℕ</mi> <mi>C</mi></msubsup><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><msub><mi>p</mi> <mi>s</mi></msub><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>≡</mo><msub><mi>c</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{ind}_\mathbb{N}^{C}(f, c_0, p_0, c_s, p_s, 0) \equiv c_0</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi mathvariant="normal">ind</mi> <mi>ℕ</mi> <mrow><mi>C</mi><mo>,</mo><mi mathvariant="normal">sec</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><msub><mi>p</mi> <mi>s</mi></msub><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>≡</mo><msub><mi>p</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{ind}_\mathbb{N}^{C, \mathrm{sec}}(f, c_0, p_0, c_s, p_s, 0) \equiv p_0</annotation></semantics></math></div> <p>and for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>:</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n:\mathbb{N}</annotation></semantics></math>,</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">ind</mi> <mi>ℕ</mi> <mi>C</mi></msubsup><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><msub><mi>p</mi> <mi>s</mi></msub><mo>,</mo><mi>s</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><msub><mi>c</mi> <mi>s</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><msubsup><mi mathvariant="normal">ind</mi> <mi>ℕ</mi> <mi>C</mi></msubsup><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><msub><mi>p</mi> <mi>s</mi></msub><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo>,</mo><msubsup><mi mathvariant="normal">ind</mi> <mi>ℕ</mi> <mrow><mi>C</mi><mo>,</mo><mi mathvariant="normal">sec</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><msub><mi>p</mi> <mi>s</mi></msub><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{ind}_\mathbb{N}^{C}(f, c_0, p_0, c_s, p_s, s(n)) \equiv c_s(n, \mathrm{ind}_\mathbb{N}^{C}(f, c_0, p_0, c_s, p_s, n), \mathrm{ind}_\mathbb{N}^{C, \mathrm{sec}}(f, c_0, p_0, c_s, p_s, n))</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi mathvariant="normal">ind</mi> <mi>ℕ</mi> <mrow><mi>C</mi><mo>,</mo><mi mathvariant="normal">sec</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><msub><mi>p</mi> <mi>s</mi></msub><mo>,</mo><mi>s</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><msub><mi>p</mi> <mi>s</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><msubsup><mi mathvariant="normal">ind</mi> <mi>ℕ</mi> <mi>C</mi></msubsup><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><msub><mi>p</mi> <mi>s</mi></msub><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo>,</mo><msubsup><mi mathvariant="normal">ind</mi> <mi>ℕ</mi> <mrow><mi>C</mi><mo>,</mo><mi mathvariant="normal">sec</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><msub><mi>p</mi> <mi>s</mi></msub><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{ind}_\mathbb{N}^{C, \mathrm{sec}}(f, c_0, p_0, c_s, p_s, s(n)) \equiv p_s(n, \mathrm{ind}_\mathbb{N}^{C}(f, c_0, p_0, c_s, p_s, n), \mathrm{ind}_\mathbb{N}^{C, \mathrm{sec}}(f, c_0, p_0, c_s, p_s, n))</annotation></semantics></math></div> <p>As inference rules these are given by the following:</p> <p><strong>elimination rules</strong>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center"><mtr><mtd><mi>Γ</mi><mo>⊢</mo><mi>C</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>f</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>ℕ</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>:</mo><mi>C</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>:</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mn>0</mn></mtd></mtr> <mtr><mtd><mi>Γ</mi><mo>⊢</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>n</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>y</mi><mo>:</mo><mi>C</mi></mrow></munder><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><msub><mi>p</mi> <mi>s</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>n</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>y</mi><mo>:</mo><mi>C</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>p</mi><mo>:</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>n</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mi>s</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>s</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr></mtable><mrow><mi>Γ</mi><mo>⊢</mo><msubsup><mi mathvariant="normal">ind</mi> <mi>ℕ</mi> <mi>C</mi></msubsup><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><msub><mi>p</mi> <mi>s</mi></msub><mo stretchy="false">)</mo><mo>:</mo><mi>ℕ</mi><mo>→</mo><mi>C</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{ \begin{array}{c} \Gamma \vdash C \; \mathrm{type} \quad \Gamma \vdash f:C \to \mathbb{N} \quad \Gamma \vdash c_0:C \quad \Gamma \vdash p_0:f(c_0) =_\mathbb{N} 0 \\ \Gamma \vdash c_s:\prod_{n:\mathbb{N}} \prod_{y:C} (f(y) =_\mathbb{N} n) \to C \quad \Gamma \vdash p_s:\prod_{n:\mathbb{N}} \prod_{y:C} \prod_{p:f(y) =_\mathbb{N} n} f(c_s(n, y, p)) =_\mathbb{N} s(n) \end{array} }{\Gamma \vdash \mathrm{ind}_\mathbb{N}^{C}(f, c_0, p_0, c_s, p_s):\mathbb{N} \to C} </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center"><mtr><mtd><mi>Γ</mi><mo>⊢</mo><mi>C</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>f</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>ℕ</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>:</mo><mi>C</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>:</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mn>0</mn></mtd></mtr> <mtr><mtd><mi>Γ</mi><mo>⊢</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>n</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>y</mi><mo>:</mo><mi>C</mi></mrow></munder><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><msub><mi>p</mi> <mi>s</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>n</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>y</mi><mo>:</mo><mi>C</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>p</mi><mo>:</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>n</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mi>s</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>s</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr></mtable><mrow><mi>Γ</mi><mo>⊢</mo><msubsup><mi mathvariant="normal">ind</mi> <mi>ℕ</mi> <mrow><mi>C</mi><mo>,</mo><mi mathvariant="normal">sec</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><msub><mi>p</mi> <mi>s</mi></msub><mo stretchy="false">)</mo><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>n</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi mathvariant="normal">ind</mi> <mi>ℕ</mi> <mi>C</mi></msubsup><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><msub><mi>p</mi> <mi>s</mi></msub><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>n</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{ \begin{array}{c} \Gamma \vdash C \; \mathrm{type} \quad \Gamma \vdash f:C \to \mathbb{N} \quad \Gamma \vdash c_0:C \quad \Gamma \vdash p_0:f(c_0) =_\mathbb{N} 0 \\ \Gamma \vdash c_s:\prod_{n:\mathbb{N}} \prod_{y:C} (f(y) =_\mathbb{N} n) \to C \quad \Gamma \vdash p_s:\prod_{n:\mathbb{N}} \prod_{y:C} \prod_{p:f(y) =_\mathbb{N} n} f(c_s(n, y, p)) =_\mathbb{N} s(n) \end{array} }{\Gamma \vdash \mathrm{ind}_\mathbb{N}^{C, \mathrm{sec}}(f, c_0, p_0, c_s, p_s):\prod_{n:\mathbb{N}} f(\mathrm{ind}_\mathbb{N}^{C}(f, c_0, p_0, c_s, p_s, n)) =_\mathbb{N} n} </annotation></semantics></math></div> <p><strong>computation rules</strong>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center"><mtr><mtd><mi>Γ</mi><mo>⊢</mo><mi>C</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>f</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>ℕ</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>:</mo><mi>C</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>:</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mn>0</mn></mtd></mtr> <mtr><mtd><mi>Γ</mi><mo>⊢</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>n</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>y</mi><mo>:</mo><mi>C</mi></mrow></munder><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><msub><mi>p</mi> <mi>s</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>n</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>y</mi><mo>:</mo><mi>C</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>p</mi><mo>:</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>n</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mi>s</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>s</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr></mtable><mrow><mi>Γ</mi><mo>⊢</mo><msubsup><mi mathvariant="normal">ind</mi> <mi>ℕ</mi> <mi>C</mi></msubsup><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><msub><mi>p</mi> <mi>s</mi></msub><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>≡</mo><msub><mi>c</mi> <mn>0</mn></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{ \begin{array}{c} \Gamma \vdash C \; \mathrm{type} \quad \Gamma \vdash f:C \to \mathbb{N} \quad \Gamma \vdash c_0:C \quad \Gamma \vdash p_0:f(c_0) =_\mathbb{N} 0 \\ \Gamma \vdash c_s:\prod_{n:\mathbb{N}} \prod_{y:C} (f(y) =_\mathbb{N} n) \to C \quad \Gamma \vdash p_s:\prod_{n:\mathbb{N}} \prod_{y:C} \prod_{p:f(y) =_\mathbb{N} n} f(c_s(n, y, p)) =_\mathbb{N} s(n) \end{array} }{\Gamma \vdash \mathrm{ind}_\mathbb{N}^{C}(f, c_0, p_0, c_s, p_s, 0) \equiv c_0} </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center"><mtr><mtd><mi>Γ</mi><mo>⊢</mo><mi>C</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>f</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>ℕ</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>:</mo><mi>C</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>:</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mn>0</mn></mtd></mtr> <mtr><mtd><mi>Γ</mi><mo>⊢</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>n</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>y</mi><mo>:</mo><mi>C</mi></mrow></munder><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><msub><mi>p</mi> <mi>s</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>n</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>y</mi><mo>:</mo><mi>C</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>p</mi><mo>:</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>n</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mi>s</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>s</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr></mtable><mrow><mi>Γ</mi><mo>⊢</mo><msubsup><mi mathvariant="normal">ind</mi> <mi>ℕ</mi> <mrow><mi>C</mi><mo>,</mo><mi mathvariant="normal">sec</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><msub><mi>p</mi> <mi>s</mi></msub><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>≡</mo><msub><mi>p</mi> <mn>0</mn></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{ \begin{array}{c} \Gamma \vdash C \; \mathrm{type} \quad \Gamma \vdash f:C \to \mathbb{N} \quad \Gamma \vdash c_0:C \quad \Gamma \vdash p_0:f(c_0) =_\mathbb{N} 0 \\ \Gamma \vdash c_s:\prod_{n:\mathbb{N}} \prod_{y:C} (f(y) =_\mathbb{N} n) \to C \quad \Gamma \vdash p_s:\prod_{n:\mathbb{N}} \prod_{y:C} \prod_{p:f(y) =_\mathbb{N} n} f(c_s(n, y, p)) =_\mathbb{N} s(n) \end{array} }{\Gamma \vdash \mathrm{ind}_\mathbb{N}^{C, \mathrm{sec}}(f, c_0, p_0, c_s, p_s, 0) \equiv p_0} </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center"><mtr><mtd><mi>Γ</mi><mo>⊢</mo><mi>C</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>f</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>ℕ</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>:</mo><mi>C</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>:</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mn>0</mn></mtd></mtr> <mtr><mtd><mi>Γ</mi><mo>⊢</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>n</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>y</mi><mo>:</mo><mi>C</mi></mrow></munder><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><msub><mi>p</mi> <mi>s</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>n</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>y</mi><mo>:</mo><mi>C</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>p</mi><mo>:</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>n</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mi>s</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>s</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>n</mi><mo>:</mo><mi>ℕ</mi></mtd></mtr></mtable><mrow><mi>Γ</mi><mo>⊢</mo><msubsup><mi mathvariant="normal">ind</mi> <mi>ℕ</mi> <mi>C</mi></msubsup><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><msub><mi>p</mi> <mi>s</mi></msub><mo>,</mo><mi>s</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><msub><mi>c</mi> <mi>s</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><msubsup><mi mathvariant="normal">ind</mi> <mi>ℕ</mi> <mi>C</mi></msubsup><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><msub><mi>p</mi> <mi>s</mi></msub><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo>,</mo><msubsup><mi mathvariant="normal">ind</mi> <mi>ℕ</mi> <mrow><mi>C</mi><mo>,</mo><mi mathvariant="normal">sec</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><msub><mi>p</mi> <mi>s</mi></msub><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{ \begin{array}{c} \Gamma \vdash C \; \mathrm{type} \quad \Gamma \vdash f:C \to \mathbb{N} \quad \Gamma \vdash c_0:C \quad \Gamma \vdash p_0:f(c_0) =_\mathbb{N} 0 \\ \Gamma \vdash c_s:\prod_{n:\mathbb{N}} \prod_{y:C} (f(y) =_\mathbb{N} n) \to C \quad \Gamma \vdash p_s:\prod_{n:\mathbb{N}} \prod_{y:C} \prod_{p:f(y) =_\mathbb{N} n} f(c_s(n, y, p)) =_\mathbb{N} s(n) \quad \Gamma \vdash n:\mathbb{N} \end{array} }{\Gamma \vdash \mathrm{ind}_\mathbb{N}^{C}(f, c_0, p_0, c_s, p_s, s(n)) \equiv c_s(n, \mathrm{ind}_\mathbb{N}^{C}(f, c_0, p_0, c_s, p_s, n), \mathrm{ind}_\mathbb{N}^{C, \mathrm{sec}}(f, c_0, p_0, c_s, p_s, n))} </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center"><mtr><mtd><mi>Γ</mi><mo>⊢</mo><mi>C</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>f</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>ℕ</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>:</mo><mi>C</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>:</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mn>0</mn></mtd></mtr> <mtr><mtd><mi>Γ</mi><mo>⊢</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>n</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>y</mi><mo>:</mo><mi>C</mi></mrow></munder><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><msub><mi>p</mi> <mi>s</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>n</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>y</mi><mo>:</mo><mi>C</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>p</mi><mo>:</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>n</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mi>s</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><mi>s</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>n</mi><mo>:</mo><mi>ℕ</mi></mtd></mtr></mtable><mrow><mi>Γ</mi><mo>⊢</mo><msubsup><mi mathvariant="normal">ind</mi> <mi>ℕ</mi> <mrow><mi>C</mi><mo>,</mo><mi mathvariant="normal">sec</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><msub><mi>p</mi> <mi>s</mi></msub><mo>,</mo><mi>s</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><msub><mi>p</mi> <mi>s</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><msubsup><mi mathvariant="normal">ind</mi> <mi>ℕ</mi> <mi>C</mi></msubsup><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><msub><mi>p</mi> <mi>s</mi></msub><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo>,</mo><msubsup><mi mathvariant="normal">ind</mi> <mi>ℕ</mi> <mrow><mi>C</mi><mo>,</mo><mi mathvariant="normal">sec</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><msub><mi>c</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>c</mi> <mi>s</mi></msub><mo>,</mo><msub><mi>p</mi> <mi>s</mi></msub><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{ \begin{array}{c} \Gamma \vdash C \; \mathrm{type} \quad \Gamma \vdash f:C \to \mathbb{N} \quad \Gamma \vdash c_0:C \quad \Gamma \vdash p_0:f(c_0) =_\mathbb{N} 0 \\ \Gamma \vdash c_s:\prod_{n:\mathbb{N}} \prod_{y:C} (f(y) =_\mathbb{N} n) \to C \quad \Gamma \vdash p_s:\prod_{n:\mathbb{N}} \prod_{y:C} \prod_{p:f(y) =_\mathbb{N} n} f(c_s(n, y, p)) =_\mathbb{N} s(n) \quad \Gamma \vdash n:\mathbb{N} \end{array} }{\Gamma \vdash \mathrm{ind}_\mathbb{N}^{C, \mathrm{sec}}(f, c_0, p_0, c_s, p_s, s(n)) \equiv p_s(n, \mathrm{ind}_\mathbb{N}^{C}(f, c_0, p_0, c_s, p_s, n), \mathrm{ind}_\mathbb{N}^{C, \mathrm{sec}}(f, c_0, p_0, c_s, p_s, n))} </annotation></semantics></math></div> <p>One gets back the usual induction principle of the natural numbers type when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>≡</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>n</mi><mo>:</mo><mi>ℕ</mi></mrow></msub><mi>P</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C \equiv \sum_{n:\mathbb{N}} P(n)</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><msub><mi>π</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">f \equiv \pi_1</annotation></semantics></math> the first projection function of the dependent sum type, and one gets back the recursion principle of the natural numbers type when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>≡</mo><mi>ℕ</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">C \equiv \mathbb{N} \times 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><msub><mi>π</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">f \equiv \pi_1</annotation></semantics></math> the first projection function of the product type.</p> <h3 id="extensionality_principle_of_the_natural_numbers">Extensionality principle of the natural numbers</h3> <p>First we <a class="existingWikiWord" href="/nlab/show/inductive+definition">inductively define</a> a <a class="existingWikiWord" href="/nlab/show/binary+function">binary function</a> into the <a class="existingWikiWord" href="/nlab/show/boolean+domain">boolean domain</a> called <a class="existingWikiWord" href="/nlab/show/observational+equality">observational equality</a> of the natural numbers:</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><mspace width="thickmathspace"></mspace><mi mathvariant="normal">ctx</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><msub><mi mathvariant="normal">Eq</mi> <mi>ℕ</mi></msub><mo>:</mo><mi>ℕ</mi><mo>×</mo><mi>ℕ</mi><mo>→</mo><mi mathvariant="normal">Bit</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{Eq}_\mathbb{N}:\mathbb{N} \times \mathbb{N} \to \mathrm{Bit}}</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><mspace width="thickmathspace"></mspace><mi mathvariant="normal">ctx</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><msup><mi>δ</mi> <mrow><mn>0</mn><mo>,</mo><mn>0</mn></mrow></msup><mo>:</mo><msub><mi mathvariant="normal">Id</mi> <mi mathvariant="normal">Bit</mi></msub><mo stretchy="false">(</mo><msub><mi mathvariant="normal">Eq</mi> <mi>ℕ</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac><mspace width="2em"></mspace><mfrac><mrow><mi>Γ</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">ctx</mi></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>n</mi><mo>:</mo><mi>ℕ</mi><mo>⊢</mo><msup><mi>δ</mi> <mrow><mn>0</mn><mo>,</mo><mi>s</mi></mrow></msup><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>:</mo><msub><mi mathvariant="normal">Id</mi> <mi mathvariant="normal">Bit</mi></msub><mo stretchy="false">(</mo><msub><mi mathvariant="normal">Eq</mi> <mi>ℕ</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>s</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \delta^{0, 0}:\mathrm{Id}_\mathrm{Bit}(\mathrm{Eq}_\mathbb{N}(0, 0), 1)} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma, n:\mathbb{N} \vdash \delta^{0, s}(n):\mathrm{Id}_\mathrm{Bit}(\mathrm{Eq}_\mathbb{N}(0, s(n)), 0)}</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">ctx</mi></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>m</mi><mo>:</mo><mi>ℕ</mi><mo>⊢</mo><msup><mi>δ</mi> <mrow><mi>s</mi><mo>,</mo><mn>0</mn></mrow></msup><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mo>:</mo><msub><mi mathvariant="normal">Id</mi> <mi mathvariant="normal">Bit</mi></msub><mo stretchy="false">(</mo><msub><mi mathvariant="normal">Eq</mi> <mi>ℕ</mi></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mfrac><mspace width="2em"></mspace><mfrac><mrow><mi>Γ</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">ctx</mi></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>m</mi><mo>:</mo><mi>ℕ</mi><mo>,</mo><mi>n</mi><mo>:</mo><mi>ℕ</mi><mo>⊢</mo><msup><mi>δ</mi> <mrow><mi>s</mi><mo>,</mo><mi>s</mi></mrow></msup><mo stretchy="false">(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo>:</mo><msub><mi mathvariant="normal">Id</mi> <mi mathvariant="normal">Bit</mi></msub><mo stretchy="false">(</mo><msub><mi mathvariant="normal">Eq</mi> <mi>ℕ</mi></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mo>,</mo><mi>s</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi mathvariant="normal">Eq</mi> <mi>ℕ</mi></msub><mo stretchy="false">(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \; \mathrm{ctx}}{\Gamma, m:\mathbb{N} \vdash \delta^{s, 0}(m):\mathrm{Id}_\mathrm{Bit}(\mathrm{Eq}_\mathbb{N}(s(m), 0), 0)} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma, m:\mathbb{N}, n:\mathbb{N} \vdash \delta^{s, s}(m, n):\mathrm{Id}_\mathrm{Bit}(\mathrm{Eq}_\mathbb{N}(s(m), s(n)),\mathrm{Eq}_\mathbb{N}(m, n))}</annotation></semantics></math></div> <p>The extensionality principle of the natural numbers states that the natural numbers has <a class="existingWikiWord" href="/nlab/show/decidable+equality">decidable equality</a> given by observational equality:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">ctx</mi></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>m</mi><mo>:</mo><mi>ℕ</mi><mo>,</mo><mi>n</mi><mo>:</mo><mi>ℕ</mi><mo>⊢</mo><mi>δ</mi><mo stretchy="false">(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo>:</mo><msub><mi mathvariant="normal">Id</mi> <mi>ℕ</mi></msub><mo stretchy="false">(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi mathvariant="normal">El</mi> <mi mathvariant="normal">Bit</mi></msub><mo stretchy="false">(</mo><msub><mi mathvariant="normal">Eq</mi> <mi>ℕ</mi></msub><mo stretchy="false">(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \; \mathrm{ctx}}{\Gamma, m:\mathbb{N}, n:\mathbb{N} \vdash \delta(m, n):\mathrm{Id}_\mathbb{N}(m, n) \simeq \mathrm{El}_\mathrm{Bit}(\mathrm{Eq}_\mathbb{N}(m, n))}</annotation></semantics></math></div> <p>or equivalently</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><mspace width="thickmathspace"></mspace><mi mathvariant="normal">ctx</mi></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>m</mi><mo>:</mo><mi>ℕ</mi><mo>,</mo><mi>n</mi><mo>:</mo><mi>ℕ</mi><mo>⊢</mo><mi>δ</mi><mo stretchy="false">(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo>:</mo><msub><mi mathvariant="normal">Id</mi> <mi>ℕ</mi></msub><mo stretchy="false">(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi mathvariant="normal">Id</mi> <mi>𝟚</mi></msub><mo stretchy="false">(</mo><msub><mi mathvariant="normal">Eq</mi> <mi>ℕ</mi></msub><mo stretchy="false">(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \; \mathrm{ctx}}{\Gamma, m:\mathbb{N}, n:\mathbb{N} \vdash \delta(m, n):\mathrm{Id}_\mathbb{N}(m, n) \simeq \mathrm{Id}_\mathbb{2}(\mathrm{Eq}_\mathbb{N}(m, n), 1)}</annotation></semantics></math></div> <h3 id="large_recursion_principle">Large recursion principle</h3> <p>In <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a> presented using only a single <a class="existingWikiWord" href="/nlab/show/type">type</a> <a class="existingWikiWord" href="/nlab/show/judgment">judgment</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><annotation encoding="application/x-tex">A \; \mathrm{type}</annotation></semantics></math>, the large recursion principle requires the need for <a class="existingWikiWord" href="/nlab/show/dependent+type+theory+with+type+variables">type variables in the dependent type theory</a> (see <a href="#CTZulip">Category Theory Zulip</a>). This is because the large recursion principle is given by the following:</p> <p>Given</p> <ol> <li> <p>a type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>0</mn></msub><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><annotation encoding="application/x-tex">T_0 \; \mathrm{type}</annotation></semantics></math></p> </li> <li> <p>a family of types <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>:</mo><mi>ℕ</mi><mo>,</mo><mi>X</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mo>⊢</mo><msub><mi>T</mi> <mi>s</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><annotation encoding="application/x-tex">n:\mathbb{N}, X \; \mathrm{type} \vdash T_s(n, X) \; \mathrm{type}</annotation></semantics></math></p> </li> </ol> <p>one can construct a family of types</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>:</mo><mi>ℕ</mi><mo>⊢</mo><msubsup><mi mathvariant="normal">rec</mi> <mi>ℕ</mi> <mrow><msub><mi>T</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>T</mi> <mi>s</mi></msub></mrow></msubsup><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><annotation encoding="application/x-tex">n:\mathbb{N} \vdash \mathrm{rec}_\mathbb{N}^{T_0, T_s}(n) \; \mathrm{type}</annotation></semantics></math></div> <p>such that</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">rec</mi> <mi>ℕ</mi> <mrow><msub><mi>T</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>T</mi> <mi>s</mi></msub></mrow></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>≡</mo><msub><mi>T</mi> <mn>0</mn></msub><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><annotation encoding="application/x-tex">\mathrm{rec}_\mathbb{N}^{T_0, T_s}(0) \equiv T_0 \; \mathrm{type}</annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>:</mo><mi>ℕ</mi><mo>⊢</mo><msubsup><mi mathvariant="normal">rec</mi> <mi>ℕ</mi> <mrow><msub><mi>T</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>T</mi> <mi>s</mi></msub></mrow></msubsup><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><msub><mi>T</mi> <mi>s</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><msubsup><mi mathvariant="normal">rec</mi> <mi>ℕ</mi> <mrow><msub><mi>T</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>T</mi> <mi>s</mi></msub></mrow></msubsup><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><annotation encoding="application/x-tex">n:\mathbb{N} \vdash \mathrm{rec}_\mathbb{N}^{T_0, T_s}(s(n)) \equiv T_s(n, \mathrm{rec}_\mathbb{N}^{T_0, T_s}(n) \; \mathrm{type}</annotation></semantics></math></div> <p>Without type variables, the second requirement in the large recursion principle that we have a family of types <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>:</mo><mi>ℕ</mi><mo>,</mo><mi>X</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mo>⊢</mo><msub><mi>T</mi> <mi>s</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><annotation encoding="application/x-tex">n:\mathbb{N}, X \; \mathrm{type} \vdash T_s(n, X) \; \mathrm{type}</annotation></semantics></math> will not be possible.</p> <h2 id="properties">Properties</h2> <h3 id="general">General</h3> <p> <div class='num_remark' id='NaturalNumbersAsWType'> <h6>Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/natural+numbers+type">natural numbers type</a> as a <a class="existingWikiWord" href="/nlab/show/W-type"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>𝒲</mi> </mrow> <annotation encoding="application/x-tex">\mathcal{W}</annotation> </semantics> </math>-type</a>)</strong> <br /> The <a class="existingWikiWord" href="/nlab/show/natural+numbers+type">natural numbers type</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℕ</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mn>0</mn><mo>,</mo><mspace width="thinmathspace"></mspace><mi>succ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbb{N},\, 0,\, succ)</annotation></semantics></math> (Def. <a class="maruku-ref" href="#InferenceRules"></a>) is equivalently the <a class="existingWikiWord" href="/nlab/show/W-type"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>𝒲</mi> </mrow> <annotation encoding="application/x-tex">\mathcal{W}</annotation> </semantics> </math>-type</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>𝒲</mi><mrow><mi>c</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi></mrow></munder><mi>A</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\underset{c \colon C}{\mathcal{W}} A(c)</annotation></semantics></math> with:</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mspace width="thinmathspace"></mspace><mo>≔</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mi>succ</mi><mo stretchy="false">}</mo><mspace width="thinmathspace"></mspace><mo>≃</mo><mspace width="thinmathspace"></mspace><mo>*</mo><mo>⊔</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">C \,\coloneqq\, \{0, succ\} \,\simeq\, \ast \sqcup \ast</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><mo>≔</mo><mspace width="thinmathspace"></mspace><mo>∅</mo></mrow><annotation encoding="application/x-tex">A_0 \,\coloneqq\, \varnothing</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/empty+type">empty type</a>);</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>succ</mi></msub><mspace width="thinmathspace"></mspace><mo>≔</mo><mspace width="thinmathspace"></mspace><mo>*</mo></mrow><annotation encoding="application/x-tex">A_{succ} \,\coloneqq\, \ast</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/unit+type">unit type</a>)</p> </li> </ul> <p></p> </div> [<a href="#Martin-Löf84">Martin-Löf (1984)</a>, <a href="/nlab/files/MartinLofIntuitionisticTypeTheory.pdf#page=51">pp. 45</a>, <a href="W-type#Dybjer97">Dybjer (1997, p. 330, 333)</a>]</p> <h3 id="relation_to_the_type_of_finite_types">Relation to the type of finite types</h3> <p>The natural numbers type is equivalent to the <a class="existingWikiWord" href="/nlab/show/set+truncation">set truncation</a> of the <a class="existingWikiWord" href="/nlab/show/type+of+finite+types">type of finite types</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi><mo>≃</mo><mo stretchy="false">[</mo><mi mathvariant="normal">FinType</mi><msub><mo stretchy="false">]</mo> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{N} \simeq [\mathrm{FinType}]_0</annotation></semantics></math></div> <p>This is the type theoretic analogue of the <a class="existingWikiWord" href="/nlab/show/decategorification">decategorification</a> of the <a class="existingWikiWord" href="/nlab/show/permutation+category">permutation category</a> resulting in the set of <a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</a>.</p> <p>This gives us an alternate definition of the natural numbers as the type of finite types</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi><mo>≔</mo><mo stretchy="false">[</mo><mi mathvariant="normal">FinType</mi><msub><mo stretchy="false">]</mo> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{N} \coloneqq [\mathrm{FinType}]_0</annotation></semantics></math></div> <p>One has <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">]</mo> <mn>0</mn></msub><mo>:</mo><mi mathvariant="normal">FinType</mi><mo>→</mo><mo stretchy="false">[</mo><mi mathvariant="normal">FinType</mi><msub><mo stretchy="false">]</mo> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">[-]_0:\mathrm{FinType} \to [\mathrm{FinType}]_0</annotation></semantics></math> by the introduction rules of set truncation.</p> <p>The arithmetic operations and order relations on the natural numbers type can be defined by induction on <a class="existingWikiWord" href="/nlab/show/set+truncation">set truncation</a>:</p> <p>For all finite types <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>:</mo><mi mathvariant="normal">FinType</mi></mrow><annotation encoding="application/x-tex">A:\mathrm{FinType}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>:</mo><mi mathvariant="normal">FinType</mi></mrow><annotation encoding="application/x-tex">B:\mathrm{FinType}</annotation></semantics></math> and finite families <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi mathvariant="normal">FinType</mi></mrow><annotation encoding="application/x-tex">C:A \to \mathrm{FinType}</annotation></semantics></math>, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><msub><mo>=</mo> <mi>ℕ</mi></msub><mo stretchy="false">[</mo><mi>∅</mi><msub><mo stretchy="false">]</mo> <mn>0</mn></msub><mspace width="1em"></mspace><mn>1</mn><msub><mo>=</mo> <mi>ℕ</mi></msub><mo stretchy="false">[</mo><mi>𝟙</mi><msub><mo stretchy="false">]</mo> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">0 =_\mathbb{N} [\emptyset]_0 \quad 1 =_\mathbb{N} [\mathbb{1}]_0</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>A</mi><msub><mo stretchy="false">]</mo> <mn>0</mn></msub><mo>+</mo><mo stretchy="false">[</mo><mi>B</mi><msub><mo stretchy="false">]</mo> <mn>0</mn></msub><msub><mo>=</mo> <mi>ℕ</mi></msub><mo stretchy="false">[</mo><mi>A</mi><mo>+</mo><mi>B</mi><msub><mo stretchy="false">]</mo> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">[A]_0 + [B]_0 =_\mathbb{N} [A + B]_0</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow> <mrow><mo stretchy="false">[</mo><mi>A</mi><msub><mo stretchy="false">]</mo> <mn>0</mn></msub></mrow></munderover><mo stretchy="false">[</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><msub><mrow><mo>[</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>]</mo></mrow> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\sum_{x = 1}^{[A]_0} [C]_0(x) =_\mathbb{N} \left[\sum_{x:A} C(x)\right]_0</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>A</mi><msub><mo stretchy="false">]</mo> <mn>0</mn></msub><mo>⋅</mo><mo stretchy="false">[</mo><mi>B</mi><msub><mo stretchy="false">]</mo> <mn>0</mn></msub><msub><mo>=</mo> <mi>ℕ</mi></msub><mo stretchy="false">[</mo><mi>A</mi><mo>×</mo><mi>B</mi><msub><mo stretchy="false">]</mo> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">[A]_0 \cdot [B]_0 =_\mathbb{N} [A \times B]_0</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow> <mrow><mo stretchy="false">[</mo><mi>A</mi><msub><mo stretchy="false">]</mo> <mn>0</mn></msub></mrow></munderover><mo stretchy="false">[</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℕ</mi></msub><msub><mrow><mo>[</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>]</mo></mrow> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\prod_{x = 1}^{[A]_0} [C]_0(x) =_\mathbb{N} \left[\prod_{x:A} C(x)\right]_0</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>B</mi><msubsup><mo stretchy="false">]</mo> <mn>0</mn> <mrow><mo stretchy="false">[</mo><mi>A</mi><msub><mo stretchy="false">]</mo> <mn>0</mn></msub></mrow></msubsup><msub><mo>=</mo> <mi>ℕ</mi></msub><mo stretchy="false">[</mo><mi>A</mi><mo>→</mo><mi>B</mi><msub><mo stretchy="false">]</mo> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">[B]_0^{[A]_0} =_\mathbb{N} [A \to B]_0</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>A</mi><msub><mo stretchy="false">]</mo> <mn>0</mn></msub><msub><mo>=</mo> <mi>ℕ</mi></msub><mo stretchy="false">[</mo><mi>B</mi><msub><mo stretchy="false">]</mo> <mn>0</mn></msub><mo>≔</mo><mo stretchy="false">[</mo><mi>A</mi><mo>≃</mo><mi>B</mi><msub><mo stretchy="false">]</mo> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mspace width="thickmathspace"></mspace><mi mathvariant="normal">or</mi><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mi>A</mi><msub><mo>=</mo> <mi mathvariant="normal">FinType</mi></msub><mi>B</mi><msub><mo stretchy="false">]</mo> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">[A]_0 =_\mathbb{N} [B]_0 \coloneqq [A \simeq B]_{(-1)} \; \mathrm{or} \; [A =_\mathrm{FinType} B]_{(-1)}</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>A</mi><msub><mo stretchy="false">]</mo> <mn>0</mn></msub><mo>≤</mo><mo stretchy="false">[</mo><mi>B</mi><msub><mo stretchy="false">]</mo> <mn>0</mn></msub><mo>≔</mo><mo stretchy="false">[</mo><mi>A</mi><mo>↪</mo><mi>B</mi><msub><mo stretchy="false">]</mo> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">[A]_0 \leq [B]_0 \coloneqq [A \hookrightarrow B]_{(-1)}</annotation></semantics></math></div> <h3 id="CategoricalSemantics">Categorical semantics</h3> <p>We spell out how under the canonical <a class="existingWikiWord" href="/nlab/show/categorical+model+of+dependent+types">categorical model of dependent types</a>, the <a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a> of the natural numbers types yields a <a class="existingWikiWord" href="/nlab/show/natural+numbers+object">natural numbers object</a> together with its expected <a class="existingWikiWord" href="/nlab/show/recursion">recursion</a> and <a class="existingWikiWord" href="/nlab/show/induction">induction</a> principle.</p> <p><br /></p> <p>Throughout, we consider an ambient <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> (e.g. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathcal{C} =</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Set">Set</a>) and write</p> <div class="maruku-equation" id="eq:CategoryOfFAlgebras"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mi>Alg</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mi>underlying</mi></mover><mi>𝒞</mi></mrow><annotation encoding="application/x-tex"> F Alg(\mathcal{C}) \xrightarrow{underlying} \mathcal{C} </annotation></semantics></math></div> <p>for the category of <a class="existingWikiWord" href="/nlab/show/algebra+over+an+endofunctor">algebras over the endofunctor</a> <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> <a class="maruku-eqref" href="#eq:TheEndofunctor">(1)</a>.</p> <h4 id="Recursion">Recursion</h4> <p>We spell out how the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">\mathbb{N}</annotation></semantics></math> satisfies Def. <a class="maruku-ref" href="#InferenceRules"></a> is the classical <a class="existingWikiWord" href="/nlab/show/recursion+principle">recursion principle</a>.</p> <p><br /></p> <p>We begin with a simple special case of recursion (cf. Rem. <a class="maruku-ref" href="#NeedForDependentRecursion"></a>), where not only the underlying type but also its successor-map is independent of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">\mathbb{N}</annotation></semantics></math> (we come to the general form of recursion further <a href="#RecursionGenerally">below</a>).</p> <p>So consider any <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>-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>D</mi><mo>,</mo><mo stretchy="false">(</mo><msub><mn>0</mn> <mi>D</mi></msub><mo>,</mo><msub><mi>succ</mi> <mi>D</mi></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mi>F</mi><mi>Alg</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\big(D, (0_D, succ_D)\big) \,\in\, F Alg(\mathcal{C})</annotation></semantics></math> <a class="maruku-eqref" href="#eq:CategoryOfFAlgebras">(2)</a>, hence an <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">D \in \mathcal{C}</annotation></semantics></math> equipped with a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mn>0</mn> <mi>D</mi></msub><mo lspace="verythinmathspace">:</mo><mo>*</mo><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex"> 0_D \colon * \to D </annotation></semantics></math></div> <p>and a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>succ</mi> <mi>D</mi></msub><mo lspace="verythinmathspace">:</mo><mi>D</mi><mo>→</mo><mi>D</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> succ_D \colon D \to D \,. </annotation></semantics></math></div> <p>By <a class="existingWikiWord" href="/nlab/show/initial+object">initiality</a> of the <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>-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">\mathbb{N}</annotation></semantics></math>, there is then a (unique) morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>rec</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>ℕ</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex"> rec \,\colon\, \mathbb{N} \to D </annotation></semantics></math></div> <p>such that the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commutes</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo>*</mo></mtd> <mtd><mover><mo>⟶</mo><mn>0</mn></mover></mtd> <mtd><mi>ℕ</mi></mtd> <mtd><mover><mo>⟶</mo><mi>succ</mi></mover></mtd> <mtd><mi>ℕ</mi></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><mpadded width="0"><mrow><msup><mo></mo><mpadded width="0"><mi>rec</mi></mpadded></msup></mrow></mpadded></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><mpadded width="0"><mrow><msup><mo></mo><mpadded width="0"><mi>rec</mi></mpadded></msup></mrow></mpadded></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mn>0</mn> <mi>D</mi></msub></mrow></munder></mtd> <mtd><mi>D</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>succ</mi> <mi>D</mi></msub></mrow></munder></mtd> <mtd><mi>D</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ * &\stackrel{0}{\longrightarrow}& \mathbb{N} &\stackrel{succ}{\longrightarrow}& \mathbb{N} \\ \big\downarrow && \big\downarrow\mathrlap{^\mathrlap{rec}} && \big\downarrow\mathrlap{^\mathrlap{rec}} \\ * &\underset{0_D}{\longrightarrow}& D &\underset{succ_D}{\longrightarrow}& D } </annotation></semantics></math></div> <p>This means precisely that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>rec</mi></mrow><annotation encoding="application/x-tex">rec</annotation></semantics></math> is the function <a class="existingWikiWord" href="/nlab/show/recursive+definition">defined recursively</a> by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>rec</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mn>0</mn> <mi>D</mi></msub></mrow><annotation encoding="application/x-tex"> rec(0) \;=\; 0_D </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation" id="eq:FormulaForNonDependentRecursion"><span class="maruku-eq-number">(3)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>rec</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>succ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi>succ</mi> <mi>D</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>rec</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> rec\big(succ(n)\big) \;=\; succ_D\big(rec(n)\big) \,. </annotation></semantics></math></div> <p id="RecursionGenerally"> More generally, consider an <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>-algebra in the <a class="existingWikiWord" href="/nlab/show/slice+category">slice</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>ℕ</mi><mo>,</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>succ</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">\big(\mathbb{N}, (0,succ)\big)</annotation></semantics></math>, but with the <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> slice object assumed (dropping also this assumption leads to the fully general notion of induction further <a href="#Induction">below</a>) to be independent of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">\mathbb{N}</annotation></semantics></math>, hence of the form</p> <div class="maruku-equation" id="eq:AssumingUnderlyingScliceObjectToBeIndependent"><span class="maruku-eq-number">(4)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>[</mo><mrow><mtable><mtr><mtd><mi>ℕ</mi><mo>×</mo><mi>D</mi></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><mpadded width="0"><mrow><msup><mo></mo><mrow><msub><mi>pr</mi> <mi>ℕ</mi></msub></mrow></msup></mrow></mpadded></mtd></mtr> <mtr><mtd><mi>ℕ</mi></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>]</mo></mrow><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>ℕ</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \left[ \array{ \mathbb{N} \times D \\ \big\downarrow\mathrlap{^{pr_{\mathbb{N}}}} \\ 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minsize="1.2em">(</mo><mi>ℕ</mi><mo>,</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi mathvariant="normal">succ</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">\big(\mathbb{N},(0,\mathrm{succ})\big)</annotation></semantics></math> being the <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mi>Alg</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F Alg(\mathcal{C})</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/identity+morphism">identity morphism</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>ℕ</mi><mo>,</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi mathvariant="normal">succ</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">\big(\mathbb{N},(0,\mathrm{succ})\big)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a> in the <a class="existingWikiWord" href="/nlab/show/slice+category">slice category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mi>Alg</mi><mo stretchy="false">(</mo><mi>𝒞</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>ℕ</mi><mo>,</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>succ</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">F Alg(\mathcal{C})_{/\big(\mathbb{N}, 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y="70.560749"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#rVIxdNn19xslb0jlaGuzz_8TdPs=-glyph14-1" x="190.876091" y="68.828571"></use> <use xlink:href="#rVIxdNn19xslb0jlaGuzz_8TdPs=-glyph14-2" x="193.784696" y="67.313033"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#rVIxdNn19xslb0jlaGuzz_8TdPs=-glyph18-1" x="198.140624" y="66.548169"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#rVIxdNn19xslb0jlaGuzz_8TdPs=-glyph15-1" x="203.405212" y="62.300233"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#rVIxdNn19xslb0jlaGuzz_8TdPs=-glyph14-3" x="206.729589" y="60.568055"></use> <use xlink:href="#rVIxdNn19xslb0jlaGuzz_8TdPs=-glyph14-4" x="209.679725" y="59.030877"></use> <use xlink:href="#rVIxdNn19xslb0jlaGuzz_8TdPs=-glyph14-5" x="213.834976" y="56.865771"></use> <use xlink:href="#rVIxdNn19xslb0jlaGuzz_8TdPs=-glyph14-5" x="217.158895" y="55.133832"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#rVIxdNn19xslb0jlaGuzz_8TdPs=-glyph18-1" x="221.099492" y="54.585377"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#rVIxdNn19xslb0jlaGuzz_8TdPs=-glyph14-8" x="226.36408" y="50.33744"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#rVIxdNn19xslb0jlaGuzz_8TdPs=-glyph13-2" x="225.220406" y="41.043184"></use> </g> </g> </svg> <p>Here the <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commutativity</a> of the top square means equivalently 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">rec</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>=</mo><mspace width="thinmathspace"></mspace><msub><mn>0</mn> <mi>D</mi></msub></mrow><annotation encoding="application/x-tex"> \mathrm{rec}(0) \,=\, 0_D </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation" id="eq:FormulaForDependentRecursion"><span class="maruku-eq-number">(5)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">rec</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi mathvariant="normal">succ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi mathvariant="normal">succ</mi> <mi>D</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>n</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi mathvariant="normal">rec</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathrm{rec}\big( \mathrm{succ}(n) \big) \;=\; \mathrm{succ}_D\big(n,\, \mathrm{rec}(n)\big) \,. </annotation></semantics></math></div> <p> <div class='num_remark' id='NeedForDependentRecursion'> <h6>Remark</h6> <p>(<strong>the need for dependent recursion</strong> [<a href="#Paulin-Mohring93">cf. Paulin-Mohring (1993, p. 330)</a>]) <br /> The appearance of the argument “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>” on the right of <a class="maruku-eqref" href="#eq:FormulaForDependentRecursion">(5)</a> – in contrast to formula <a class="maruku-eqref" href="#eq:FormulaForNonDependentRecursion">(3)</a> for non-dependent recursion – means (in view of the argument “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>succ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">succ(n)</annotation></semantics></math>” on the left) that the recursor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>succ</mi> <mi>D</mi></msub></mrow><annotation encoding="application/x-tex">succ_D</annotation></semantics></math> has access to the <em><a class="existingWikiWord" href="/nlab/show/predecessor">predecessor</a></em> <a class="existingWikiWord" href="/nlab/show/partial+function">partial function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>pred</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>succ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>↦</mo><mspace width="thinmathspace"></mspace><mi>n</mi></mrow><annotation encoding="application/x-tex">pred \,\colon\,succ(n) \,\mapsto\, n</annotation></semantics></math>. This is necessary in order to express all computable functions on the natural numbers inductively and hence explains the need for the <a class="existingWikiWord" href="/nlab/show/dependent+type">dependently typed</a> recursion principle <a class="maruku-eqref" href="#eq:AssumingUnderlyingScliceObjectToBeIndependent">(4)</a></p> </div> </p> <h4 id="Induction">Induction</h4> <p>Dropping the above constraint <a class="maruku-eqref" href="#eq:AssumingUnderlyingScliceObjectToBeIndependent">(4)</a> on the dependent <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>-algebra, we spell out in detail how the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">\mathbb{N}</annotation></semantics></math> satisfied Def. <a class="maruku-ref" href="#InferenceRules"></a> is the classical <a class="existingWikiWord" href="/nlab/show/induction+principle">induction principle</a>.</p> <p>That principle says informally that if a <a class="existingWikiWord" href="/nlab/show/proposition">proposition</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> depending on the natural numbers is true at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n = 0</annotation></semantics></math> and such that if it is true for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> then it is true for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math>, then it is true for all natural numbers.</p> <p>Here is how this is formalized in type theory and then <a class="existingWikiWord" href="/nlab/show/categorical+semantics">interpreted</a> in some suitable ambient category <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">\mathcal{C}</annotation></semantics></math>.</p> <p>First of all, that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> is a proposition depending on the natural numbers means that it is a <a class="existingWikiWord" href="/nlab/show/dependent+type">dependent type</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>n</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>ℕ</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊢</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>P</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>Type</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> n \,\colon\, \mathbb{N} \;\;\vdash\;\; P(n) \,\colon\, Type \,. </annotation></semantics></math></div> <p>The categorical interpretation of this is by a <a class="existingWikiWord" href="/nlab/show/display+map">display map</a></p> <div class="maruku-equation" id="eq:DisplayMapInterpretingNDependentType"><span class="maruku-eq-number">(6)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>P</mi></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><mpadded width="0"><mrow><msup><mo></mo><mrow><msub><mi>π</mi> <mi>P</mi></msub></mrow></msup></mrow></mpadded></mtd></mtr> <mtr><mtd><mi>ℕ</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ P \\ \big\downarrow\mathrlap{^{\pi_P}} \\ \mathbb{N} } </annotation></semantics></math></div> <p>in the given category <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">\mathcal{C}</annotation></semantics></math>.</p> <p id="DiagramInterpretingInductionPrinciple"> With this, the <a 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x="150.818828" y="71.117667"></use> <use xlink:href="#JRUx4CTiaqvLLGaYGmAHiCOeAZg=-glyph12-4" x="153.642007" y="69.341046"></use> <use xlink:href="#JRUx4CTiaqvLLGaYGmAHiCOeAZg=-glyph12-5" x="157.61844" y="66.838685"></use> <use xlink:href="#JRUx4CTiaqvLLGaYGmAHiCOeAZg=-glyph12-5" x="160.799317" y="64.836966"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JRUx4CTiaqvLLGaYGmAHiCOeAZg=-glyph13-1" x="164.692396" y="63.966954"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JRUx4CTiaqvLLGaYGmAHiCOeAZg=-glyph12-6" x="169.160411" y="59.575347"></use> </g> </g> </svg> <p>We now unwind again how this comes about and what it all means:</p> <p>First, the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> holds at 0 means that there is a (<a class="existingWikiWord" href="/nlab/show/proof">proof</a>-)<a class="existingWikiWord" href="/nlab/show/term">term</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>⊢</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mn>0</mn> <mi>P</mi></msub><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>P</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \vdash \;\; 0_P \,\colon\, P(0) \,. </annotation></semantics></math></div> <p>In the <a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a> the <a class="existingWikiWord" href="/nlab/show/substitution">substitution</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> for 0 that gives <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(0)</annotation></semantics></math> is given by the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of the given display map <a class="maruku-eqref" href="#eq:DisplayMapInterpretingNDependentType">(6)</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mn>0</mn> <mo>*</mo></msup><mi>P</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>P</mi></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><munder><mo>⟶</mo><mn>0</mn></munder></mtd> <mtd><mi>ℕ</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ 0^* P &\longrightarrow& P \\ \big\downarrow && \big\downarrow \\ * &\underset{0}{\longrightarrow}& \mathbb{N} } </annotation></semantics></math></div> <p>and the <a class="existingWikiWord" href="/nlab/show/term">term</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mn>0</mn> <mi>P</mi></msub></mrow><annotation encoding="application/x-tex">0_P</annotation></semantics></math> is interpreted as a <a class="existingWikiWord" href="/nlab/show/section">section</a> of the resulting fibration over the terminal object</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo>*</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>p</mi> <mn>0</mn></msub></mrow></mover></mtd> <mtd><msup><mn>0</mn> <mo>*</mo></msup><mi>P</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>P</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd> <mtd><munder><mo>⟶</mo><mn>0</mn></munder></mtd> <mtd><mi>ℕ</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ * &\overset{p_0}{\longrightarrow}& 0^* P & \longrightarrow & P \\ &\searrow& \big\downarrow && \big\downarrow \\ && * &\underset{0}{\longrightarrow}& \mathbb{N} } \,. </annotation></semantics></math></div> <p>But by the defining <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a>, this is equivalently just a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo>*</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>p</mi> <mn>0</mn></msub></mrow></mover></mtd> <mtd><mi>P</mi></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><munder><mo>⟶</mo><mn>0</mn></munder></mtd> <mtd><mi>ℕ</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ * &\stackrel{p_0}{\longrightarrow}& P \\ \big\downarrow && \big\downarrow \\ * &\underset{0}{\longrightarrow}& \mathbb{N} } \,. </annotation></semantics></math></div> <p>Next the induction step. Formally it says that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> there is an <a class="existingWikiWord" href="/nlab/show/implication">implication</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>succ</mi> <mi>P</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>P</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mi>P</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">succ_P(n) \,\colon\, P(n) \to P(n+1)</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊢</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>succ</mi> <mi>P</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>P</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mi>P</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> n \in \mathbb{N} \;\;\;\vdash\;\;\; succ_P(n) \,\colon\, P(n) \to P(n+1) \,. </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a> of the <a class="existingWikiWord" href="/nlab/show/substitution">substitution</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> is given by the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>P</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mo>*</mo><mo>⊔</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>≔</mo></mtd> <mtd><msup><mi>s</mi> <mo>*</mo></msup><mi>P</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>P</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mi>ℕ</mi></mtd> <mtd><munder><mo>⟶</mo><mi>s</mi></munder></mtd> <mtd><mi>ℕ</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ P\big(\ast \sqcup (-)\big) \coloneqq & s^*P &\longrightarrow& P \\ & \big\downarrow && \big\downarrow \\ & \mathbb{N} &\underset{s}{\longrightarrow}& \mathbb{N} } </annotation></semantics></math></div> <p>and the interpretation of the implication term <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>succ</mi> <mi>P</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">succ_P(n)</annotation></semantics></math> is as a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><msup><mi>s</mi> <mo>*</mo></msup><mi>P</mi></mrow><annotation encoding="application/x-tex">P \to s^* P</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>ℕ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_{/\mathbb{N}}</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>P</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>p</mi> <mi>s</mi></msub></mrow></mover></mtd> <mtd><msup><mi>s</mi> <mo>*</mo></msup><mi>P</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>P</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>ℕ</mi></mtd> <mtd><mover><mo>⟶</mo><mi>s</mi></mover></mtd> <mtd><mi>ℕ</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ P & \overset{p_s}{\longrightarrow} & s^*P &\longrightarrow& P \\ &\searrow & \big\downarrow && \big\downarrow \\ && \mathbb{N} &\overset{s}{\longrightarrow}& \mathbb{N} } \,. </annotation></semantics></math></div> <p>Again by the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the pullback this is equivalently a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>P</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>succ</mi> <mi>P</mi></msub></mrow></mover></mtd> <mtd><mi>P</mi></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd></mtr> <mtr><mtd><mi>ℕ</mi></mtd> <mtd><munder><mo>⟶</mo><mi>s</mi></munder></mtd> <mtd><mi>ℕ</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ P &\overset{succ_P}{\longrightarrow}& P \\ \big\downarrow && \big\downarrow \\ \mathbb{N} &\underset{s}{\longrightarrow}& \mathbb{N} } \,. </annotation></semantics></math></div> <p>In summary this shows that</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> being a proposition depending on natural numbers which holds at 0 and which holds at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math> if it holds at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></li> </ul> <p>is interpreted precisely as an <a class="existingWikiWord" href="/nlab/show/algebra+for+an+endofunctor">endofunctor-algebra homomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>P</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>ℕ</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ P \\ \downarrow \\ \mathbb{N} } \,. </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/endofunctor">endofunctor</a> <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> <a class="maruku-eqref" href="#eq:TheEndofunctor">(1)</a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/induction+principle">induction principle</a> is supposed to deduce from this that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> holds for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>, hence that there is a proof <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ind</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>P</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ind(n) \colon P(n)</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>n</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>ℕ</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊢</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>ind</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>P</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> n \,\colon\, \mathbb{N} \;\;\;\vdash\;\;\; ind(n) \,\colon\, P(n) \,. </annotation></semantics></math></div> <p>The categorical interpretation of this is as a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>ℕ</mi><mo>→</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">p \,\colon\, \mathbb{N} \to P</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>ℕ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_{/\mathbb{N}}</annotation></semantics></math>. The existence of this is indeed exactly what the interpretation of the elimination rule (Def. <a class="maruku-ref" href="#InferenceRules"></a>) gives exactly what the initiality of the <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>-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">\mathbb{N}</annotation></semantics></math> gives.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</a>, <a class="existingWikiWord" href="/nlab/show/natural+numbers+object">natural numbers object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/decimal+numeral+representation+of+the+natural+numbers">decimal numeral representation of the natural numbers</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dependent+sequence+type">dependent sequence type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/type+of+finite+types">type of finite types</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dependent+type+theory+with+type+variables">dependent type theory with type variables</a></p> </li> </ul> <h2 id="references">References</h2> <p>Original articles with emphasis on the nature of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">\mathbb{N}</annotation></semantics></math> as an <a class="existingWikiWord" href="/nlab/show/inductive+type">inductive type</a>:</p> <ul> <li id="Martin-Löf84"> <p><a class="existingWikiWord" href="/nlab/show/Per+Martin-L%C3%B6f">Per Martin-Löf</a> (notes by <a class="existingWikiWord" href="/nlab/show/Giovanni+Sambin">Giovanni Sambin</a>), <a href="/nlab/files/MartinLofIntuitionisticTypeTheory.pdf#page=44">pp. 38</a> of: <em>Intuitionistic type theory</em>, Lecture notes Padua 1984, Bibliopolis, Napoli (1984) [<a href="https://archive-pml.github.io/martin-lof/pdfs/Bibliopolis-Book-retypeset-1984.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/MartinLofIntuitionisticTypeTheory.pdf" title="pdf">pdf</a>]</p> </li> <li id="CoquandPaulin90"> <p><a class="existingWikiWord" href="/nlab/show/Thierry+Coquand">Thierry Coquand</a>, <a class="existingWikiWord" href="/nlab/show/Christine+Paulin">Christine Paulin</a>, p. 52-53 in: <em>Inductively defined types</em>, COLOG-88 Lecture Notes in Computer Science <strong>417</strong>, Springer (1990) 50-66 [<a href="https://doi.org/10.1007/3-540-52335-9_47">doi:10.1007/3-540-52335-9_47</a>]</p> </li> <li id="Paulin-Mohring93"> <p><a class="existingWikiWord" href="/nlab/show/Christine+Paulin-Mohring">Christine Paulin-Mohring</a>, §1.3 in: <em>Inductive definitions in the system Coq – Rules and Properties</em>, in: <em>Typed Lambda Calculi and Applications</em> TLCA 1993, Lecture Notes in Computer Science <strong>664</strong> Springer (1993) [<a href="https://doi.org/10.1007/BFb0037116">doi:10.1007/BFb0037116</a>]</p> </li> <li id="Dybjer94"> <p><a class="existingWikiWord" href="/nlab/show/Peter+Dybjer">Peter Dybjer</a>, §3 in: <em>Inductive families</em>, Formal Aspects of Computing <strong>6</strong> (1994) 440–465 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1007/BF01211308">doi:10.1007/BF01211308</a>, <a href="https://doi.org/10.1007/BF01211308">doi:10.1007/BF01211308</a>, <a href="http://www.cse.chalmers.se/~peterd/papers/Inductive_Families.pdf">pdf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> </ul> <p>The syntax in <a class="existingWikiWord" href="/nlab/show/Coq">Coq</a>:</p> <ul> <li id="Paulin-Mohring14"><a class="existingWikiWord" href="/nlab/show/Christine+Paulin-Mohring">Christine Paulin-Mohring</a>, p. 6 in: <em>Introduction to the Calculus of Inductive Constructions</em>, contribution to: <em>Vienna Summer of Logic</em> (2014) [<a href="https://hal.inria.fr/hal-01094195">hal:01094195</a>, <a href="https://hal.inria.fr/hal-01094195/document">pdf</a>, <a href="https://easychair.org/smart-program/VSL2014/APPA-invited-slides-6.pdf">pdf slides</a>]</li> </ul> <p>See also:</p> <ul> <li id="Pfenning"><a class="existingWikiWord" href="/nlab/show/Frank+Pfenning">Frank Pfenning</a>, <em>Lecture notes on natural numbers</em> (2009) [<a href="http://www.cs.cmu.edu/~fp/courses/15317-f09/lectures/06-nat.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Pfenning-NaturalNumbersType.pdf" title="pdf">pdf</a>]</li> </ul> <p>Discussion in a context of <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> and in view of <a class="existingWikiWord" href="/nlab/show/higher+inductive+types">higher inductive types</a>:</p> <ul> <li id="UFP13"> <p><a class="existingWikiWord" href="/nlab/show/Univalent+Foundations+Project">Univalent Foundations Project</a>, §1.9 in: <em><a class="existingWikiWord" href="/nlab/show/Homotopy+Type+Theory+--+Univalent+Foundations+of+Mathematics">Homotopy Type Theory – Univalent Foundations of Mathematics</a></em> (2013) [<a href="http://homotopytypetheory.org/book/">web</a>, <a href="http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Egbert+Rijke">Egbert Rijke</a>, §3 in: <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>)</p> </li> <li id="Söhnen18"> <p>Kajetan Söhnen, §2.4.5 in: <em>Higher Inductive Types in Homotopy Type Theory</em>, Munich (2018) [<a href="https://www.math.lmu.de/~petrakis/Soehnen.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Soehnen-HigherInductiveTypes.pdf" title="pdf">pdf</a>]</p> </li> <li id="LumsdaineShulman17"> <p><a class="existingWikiWord" href="/nlab/show/Peter+LeFanu+Lumsdaine">Peter LeFanu Lumsdaine</a>, <a class="existingWikiWord" href="/nlab/show/Mike+Shulman">Mike Shulman</a>, <em>Semantics of higher inductive types</em>, Math. Proc. Camb. Phil. Soc. <strong>169</strong> (2020) 159-208 [<a href="https://arxiv.org/abs/1705.07088">arXiv:1705.07088</a>, talk slides <a href="http://home.sandiego.edu/~shulman/papers/cellcxs.pdf">pdf</a>, <a href="https://doi.org/10.1017/S030500411900015X">doi:10.1017/S030500411900015X</a>]</p> </li> </ul> <p>Equivalence to binary presentations:</p> <ul> <li> <p><span class="newWikiWord">Nicolas Magaud<a href="/nlab/new/Nicolas+Magaud">?</a></span>, <span class="newWikiWord">Yves Bertot<a href="/nlab/new/Yves+Bertot">?</a></span>, <em>Changing Data Structures in Type Theory: A Study of Natural Numbers</em>, in <em>Types for Proofs and Programs. TYPES 2000</em>, Lecture Notes in Computer Science <strong>2277</strong> [<a href="https://doi.org/10.1007/3-540-45842-5_12">doi:10.1007/3-540-45842-5_12</a>, <a href="https://dpt-info.u-strasbg.fr/~magaud/papers/types2000-nmagaud.pdf">pdf</a>]</p> </li> <li> <p><span class="newWikiWord">Nicolas Magaud<a href="/nlab/new/Nicolas+Magaud">?</a></span>, <em>Changing Data Representation within the Coq</em>, in <em>Theorem Proving in Higher Order Logics. TPHOLs 2003</em>, Lecture Notes in Computer Science <strong>2758</strong> [<a href="https://doi.org/10.1007/10930755_6">doi:10.1007/10930755_6</a>]</p> </li> </ul> <p>Some discussion about the large recursion principle of the natural numbers type in <a class="existingWikiWord" href="/nlab/show/dependent+type+theory+with+type+variables">dependent type theory with type variables</a> occurs in:</p> <ul> <li id="CTZulip"><em>Dependent Type Theory vs Polymorphic Type Theory</em>, Category Theory Zulip (<a href="https://categorytheory.zulipchat.com/#narrow/stream/229199-learning.3A-questions/topic/Dependent.20Type.20Theory.20vs.20Polymorphic.20Type.20Theory">web</a>)</li> </ul> <p>That one can construct the <a class="existingWikiWord" href="/nlab/show/natural+numbers+type">natural numbers type</a> from the integers type can be found in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Christian+Sattler">Christian Sattler</a>, <em>Natural numbers from integers</em> (<a href="https://www.cse.chalmers.se/~sattler/docs/naturals.pdf">pdf</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 17, 2024 at 03:24:12. 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