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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='#rules_for_types_of_lists'>Rules for types of lists</a></li> </ul> <li><a href='#see_also'>See also</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="Idea">Idea</h2> <p>In <a class="existingWikiWord" href="/nlab/show/mathematics">mathematics</a> and specifically in <a class="existingWikiWord" href="/nlab/show/formal+logic">formal logic</a>, by a “list” one means the fairly evident formalization of the colloquial notion of a “list of elements”:</p> <p>If a <a class="existingWikiWord" href="/nlab/show/set">set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> of admissible elements is given — also called an “<a class="existingWikiWord" href="/nlab/show/alphabet">alphabet</a>”, in this context — then a list of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-elements is a <a class="existingWikiWord" href="/nlab/show/tuple">tuple</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>e</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>e</mi> <mn>2</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>e</mi> <mi>ℓ</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(e_1, e_2, \cdots, e_\ell)</annotation></semantics></math> of elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>i</mi></msub><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mi>E</mi></mrow><annotation encoding="application/x-tex">e_i \,\in\, E</annotation></semantics></math>, of any length <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℓ</mi><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">\ell \,\in\, \mathbb{N}</annotation></semantics></math> — also called a <em><a class="existingWikiWord" href="/nlab/show/word">word</a></em> in the given alphabet.</p> <p>Typically one will write “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>List</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">List(E)</annotation></semantics></math>” or similar for the <a class="existingWikiWord" href="/nlab/show/set">set</a> of all lists with entries in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>.</p> <p>For instance, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> the (<a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> set of) a <a class="existingWikiWord" href="/nlab/show/field">field</a>, the component-expressions of <a class="existingWikiWord" href="/nlab/show/vectors">vectors</a> in the <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">k^n</annotation></semantics></math> may be thought of as lists of length <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> with elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>. However, doing so means to disregard the <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>-<a class="existingWikiWord" href="/nlab/show/structure">structure</a> on the collection of all such lists.</p> <p>Instead, the notion of <em>list</em> is a <a class="existingWikiWord" href="/nlab/show/concept+with+an+attitude">concept with an attitude</a>: While lists are just <a class="existingWikiWord" href="/nlab/show/tuples">tuples</a> of any length, calling them lists indicates that one intends to consider the operation of <em><a class="existingWikiWord" href="/nlab/show/concatenation">concatenation</a></em> of lists to larger lists, in particular the operations of appending an element to a list.</p> <p>It is evident that the operation of concatenation makes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>List</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">List(E)</annotation></semantics></math> a <em><a class="existingWikiWord" href="/nlab/show/monoid">monoid</a></em> (the <a class="existingWikiWord" href="/nlab/show/neutral+element">neutral element</a> under concatenation is the <a class="existingWikiWord" href="/nlab/show/empty+list">empty list</a>, i.e. the unique list of length <a class="existingWikiWord" href="/nlab/show/zero">zero</a>). In fact, a moment of reflection shows that, as such, <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>List</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>,</mo><mi>conc</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">\big(List(E), conc\big)</annotation></semantics></math> is the <em><a class="existingWikiWord" href="/nlab/show/free+monoid">free monoid</a></em> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>.</p> <p>Therefore, in much of the mathematical literature, lists are understood as <a class="existingWikiWord" href="/nlab/show/free+monoids">free monoids</a>.</p> <p>Specifically in <a class="existingWikiWord" href="/nlab/show/computer+science">computer science</a> one commonly deals with the corresponding notion of the <a class="existingWikiWord" href="/nlab/show/data+type">data type</a> of lists (with entries in a prescribed data type), which serve as a basic and common kind of <span class="newWikiWord">data structure<a href="/nlab/new/data+structure">?</a></span>. In <a class="existingWikiWord" href="/nlab/show/programming+languages">programming languages</a> supporting something like a <a class="existingWikiWord" href="/nlab/show/calculus+of+constructions">calculus of constructions</a>, this <a class="existingWikiWord" href="/nlab/show/data+type">data type</a>, essentially with the <a class="existingWikiWord" href="/nlab/show/free+monoid">free monoid</a>-<a class="existingWikiWord" href="/nlab/show/structure">structure</a> as above, may be defined as an <em><a class="existingWikiWord" href="/nlab/show/inductive+type">inductive type</a></em> in an evident way (made explicit <a href="#Definition">below</a>).</p> <p>On the other hand, in <a class="existingWikiWord" href="/nlab/show/dependent+type+theories">dependent type theories</a> which have a <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>-interpretation — in that they do not verify <a class="existingWikiWord" href="/nlab/show/uniqueness+of+identity+proofs">uniqueness of identity proofs</a> (or “<a class="existingWikiWord" href="/nlab/show/axiom+K+%28type+theory%29">axiom K</a>”) — one may want to distinguish between lists and <a class="existingWikiWord" href="/nlab/show/free+monoids">free monoids</a>:</p> <p>Because, in such theories it may happen that the given <a class="existingWikiWord" href="/nlab/show/alphabet">alphabet</a> <a class="existingWikiWord" href="/nlab/show/type">type</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> is not actually a <a class="existingWikiWord" href="/nlab/show/set">set</a> but a higher <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a>, in that it is not <a class="existingWikiWord" href="/nlab/show/0-truncated">0-truncated</a>. In this case it still makes good sense to speak of <em>lists</em> of elements (<a class="existingWikiWord" href="/nlab/show/terms">terms</a>) of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>, only that now the corresponding <a class="existingWikiWord" href="/nlab/show/type">type</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>List</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">List(E)</annotation></semantics></math> is itself not <a class="existingWikiWord" href="/nlab/show/0-truncated">0-truncated</a>. But since the term “<a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>” carries with it a connotation of being <a class="existingWikiWord" href="/nlab/show/0-truncated">0-truncated</a>, one may no longer want to call this the <em><a class="existingWikiWord" href="/nlab/show/free+monoid">free monoid</a></em> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>. It is, of course, still a free monoid in the proper <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebraic</a> sense (cf. <em><a class="existingWikiWord" href="/nlab/show/monoid+in+a+monoidal+%28infinity%2C1%29-category">monoid in a monoidal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-category</a></em>).</p> <h2 id="Definition">Definition</h2> <p>The type of lists on a type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is defined as the <a class="existingWikiWord" href="/nlab/show/dependent+sum+type">dependent sum type</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">List</mi><mo stretchy="false">(</mo><mi>A</mi><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 mathvariant="normal">Fin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\mathrm{List}(A) \coloneqq \sum_{n:\mathbb{N}} \mathrm{Fin}(n) \to A</annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">Fin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{Fin}(n)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/finite+set">finite set</a> with <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> elements. This shows that lists are just <a class="existingWikiWord" href="/nlab/show/tuples">tuples</a>.</p> <h3 id="rules_for_types_of_lists">Rules for types of lists</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+product+types">dependent product types</a> exist in the type theory, the type of lists on a type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/generators">generators</a> is the <a class="existingWikiWord" href="/nlab/show/inductive+type">inductive type</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{List}(A)</annotation></semantics></math> generated by an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">nil</mi><mo>:</mo><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{nil}:\mathrm{List}(A)</annotation></semantics></math> and a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">cons</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{cons}:A \to \mathrm{List}(A) \to \mathrm{List}(A)</annotation></semantics></math>:</p> <p>Formation rules for the type of lists:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{List}(A) \; \mathrm{type}}</annotation></semantics></math></div> <p>Introduction rules for the type of lists:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><mi mathvariant="normal">nil</mi><mo>:</mo><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></mfrac><mspace width="2em"></mspace><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><mi mathvariant="normal">cons</mi><mo>:</mo><mi>A</mi><mo>→</mo><mo stretchy="false">(</mo><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{nil}:\mathrm{List}(A)} \qquad \frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{cons}:A \to (\mathrm{List}(A) \to \mathrm{List}(A))}</annotation></semantics></math></div> <p>Elimination rules for the type of lists:</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="left"><mtr><mtd><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><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></mtd></mtr> <mtr><mtd><mi>Γ</mi><mo>⊢</mo><msub><mi>c</mi> <mi mathvariant="normal">nil</mi></msub><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mi mathvariant="normal">nil</mi><mo stretchy="false">)</mo><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><msub><mi>c</mi> <mi mathvariant="normal">cons</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>a</mi><mo>:</mo><mi>A</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></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 mathvariant="normal">cons</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mi>Γ</mi><mo>⊢</mo><mi>g</mi><mo>:</mo><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable><mrow><mi>Γ</mi><mo>⊢</mo><msubsup><mi mathvariant="normal">ind</mi> <mrow><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow> <mi>C</mi></msubsup><mo stretchy="false">(</mo><msub><mi>c</mi> <mi mathvariant="normal">nil</mi></msub><mo>,</mo><msub><mi>c</mi> <mi mathvariant="normal">cons</mi></msub><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{ \begin{array}{l} \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:\mathrm{List}(A) \vdash C(x) \; \mathrm{type} \\ \Gamma \vdash c_\mathrm{nil}:C(\mathrm{nil}) \quad \Gamma \vdash c_\mathrm{cons}:\prod_{a:A} \prod_{x:\mathrm{List}(A))} C(x) \to C(\mathrm{cons}(a)(x)) \\ \Gamma \vdash g:\mathrm{List}(A)) \end{array} }{\Gamma \vdash \mathrm{ind}_{\mathrm{List}(A)}^C(c_\mathrm{nil}, c_\mathrm{cons}, g):C(g)} </annotation></semantics></math></div> <p>Computation rules for the type of lists:</p> <ul> <li>Judgmental computation rules</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mtable displaystyle="false" rowspacing="0.5ex" columnalign="left"><mtr><mtd><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><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></mtd></mtr> <mtr><mtd><mi>Γ</mi><mo>⊢</mo><msub><mi>c</mi> <mi mathvariant="normal">nil</mi></msub><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mi mathvariant="normal">nil</mi><mo stretchy="false">)</mo><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><msub><mi>c</mi> <mi mathvariant="normal">cons</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>a</mi><mo>:</mo><mi>A</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></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 mathvariant="normal">cons</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable><mrow><mi>Γ</mi><mo>⊢</mo><msubsup><mi mathvariant="normal">ind</mi> <mrow><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow> <mi>C</mi></msubsup><mo stretchy="false">(</mo><msub><mi>c</mi> <mi mathvariant="normal">nil</mi></msub><mo>,</mo><msub><mi>c</mi> <mi mathvariant="normal">cons</mi></msub><mo>,</mo><mi mathvariant="normal">nil</mi><mo stretchy="false">)</mo><mo>≡</mo><msub><mi>c</mi> <mi mathvariant="normal">nil</mi></msub><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mi mathvariant="normal">nil</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{ \begin{array}{l} \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:\mathrm{List}(A) \vdash C(x) \; \mathrm{type} \\ \Gamma \vdash c_\mathrm{nil}:C(\mathrm{nil}) \quad \Gamma \vdash c_\mathrm{cons}:\prod_{a:A} \prod_{x:\mathrm{List}(A))} C(x) \to C(\mathrm{cons}(a)(x)) \end{array} }{\Gamma \vdash \mathrm{ind}_{\mathrm{List}(A)}^C(c_\mathrm{nil}, c_\mathrm{cons}, \mathrm{nil}) \equiv c_\mathrm{nil}:C(\mathrm{nil})} </annotation></semantics></math></div> <p><br /></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="left"><mtr><mtd><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><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></mtd></mtr> <mtr><mtd><mi>Γ</mi><mo>⊢</mo><msub><mi>c</mi> <mi mathvariant="normal">nil</mi></msub><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mi mathvariant="normal">nil</mi><mo stretchy="false">)</mo><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><msub><mi>c</mi> <mi mathvariant="normal">cons</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>a</mi><mo>:</mo><mi>A</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></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 mathvariant="normal">cons</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mi>Γ</mi><mo>⊢</mo><mi>b</mi><mo>:</mo><mi>A</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>g</mi><mo>:</mo><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable><mrow><mi>Γ</mi><mo>⊢</mo><msubsup><mi mathvariant="normal">ind</mi> <mrow><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow> <mi>C</mi></msubsup><mo stretchy="false">(</mo><msub><mi>c</mi> <mi mathvariant="normal">nil</mi></msub><mo>,</mo><msub><mi>c</mi> <mi mathvariant="normal">cons</mi></msub><mo>,</mo><mi mathvariant="normal">cons</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><msub><mi>c</mi> <mi mathvariant="normal">cons</mi></msub><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msubsup><mi mathvariant="normal">ind</mi> <mrow><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow> <mi>C</mi></msubsup><mo stretchy="false">(</mo><msub><mi>c</mi> <mi mathvariant="normal">nil</mi></msub><mo>,</mo><msub><mi>c</mi> <mi mathvariant="normal">cons</mi></msub><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mi mathvariant="normal">cons</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{ \begin{array}{l} \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:\mathrm{List}(A) \vdash C(x) \; \mathrm{type} \\ \Gamma \vdash c_\mathrm{nil}:C(\mathrm{nil}) \quad \Gamma \vdash c_\mathrm{cons}:\prod_{a:A} \prod_{x:\mathrm{List}(A))} C(x) \to C(\mathrm{cons}(a)(x)) \\ \Gamma \vdash b:A \quad \Gamma \vdash g:\mathrm{List}(A)) \end{array} }{\Gamma \vdash \mathrm{ind}_{\mathrm{List}(A)}^C(c_\mathrm{nil}, c_\mathrm{cons}, \mathrm{cons}(b)(g)) \equiv c_\mathrm{cons}(b)(g)(\mathrm{ind}_{\mathrm{List}(A)}^C(c_\mathrm{nil}, c_\mathrm{cons}, g)):C(\mathrm{cons}(b)(g))} </annotation></semantics></math></div> <p><br /></p> <ul> <li>Typal computation rules</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mtable displaystyle="false" rowspacing="0.5ex" columnalign="left"><mtr><mtd><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><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></mtd></mtr> <mtr><mtd><mi>Γ</mi><mo>⊢</mo><msub><mi>c</mi> <mi mathvariant="normal">nil</mi></msub><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mi mathvariant="normal">nil</mi><mo stretchy="false">)</mo><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><msub><mi>c</mi> <mi mathvariant="normal">cons</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>a</mi><mo>:</mo><mi>A</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></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 mathvariant="normal">cons</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable><mrow><mi>Γ</mi><mo>⊢</mo><msubsup><mi>β</mi> <mrow><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow> <mi mathvariant="normal">nil</mi></msubsup><mo stretchy="false">(</mo><msub><mi>c</mi> <mi mathvariant="normal">nil</mi></msub><mo>,</mo><msub><mi>c</mi> <mi mathvariant="normal">cons</mi></msub><mo stretchy="false">)</mo><mo>:</mo><msub><mi mathvariant="normal">Id</mi> <mrow><mi>C</mi><mo stretchy="false">(</mo><mi mathvariant="normal">nil</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msubsup><mi mathvariant="normal">ind</mi> <mrow><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow> <mi>C</mi></msubsup><mo stretchy="false">(</mo><msub><mi>c</mi> <mi mathvariant="normal">nil</mi></msub><mo>,</mo><msub><mi>c</mi> <mi mathvariant="normal">cons</mi></msub><mo>,</mo><mi mathvariant="normal">nil</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>c</mi> <mi mathvariant="normal">nil</mi></msub><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{ \begin{array}{l} \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:\mathrm{List}(A) \vdash C(x) \; \mathrm{type} \\ \Gamma \vdash c_\mathrm{nil}:C(\mathrm{nil}) \quad \Gamma \vdash c_\mathrm{cons}:\prod_{a:A} \prod_{x:\mathrm{List}(A))} C(x) \to C(\mathrm{cons}(a)(x)) \end{array} }{\Gamma \vdash \beta_{\mathrm{List}(A)}^\mathrm{nil}(c_\mathrm{nil}, c_\mathrm{cons}):\mathrm{Id}_{C(\mathrm{nil})}(\mathrm{ind}_{\mathrm{List}(A)}^C(c_\mathrm{nil}, c_\mathrm{cons}, \mathrm{nil}), c_\mathrm{nil})} </annotation></semantics></math></div> <p><br /></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="left"><mtr><mtd><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><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></mtd></mtr> <mtr><mtd><mi>Γ</mi><mo>⊢</mo><msub><mi>c</mi> <mi mathvariant="normal">nil</mi></msub><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mi mathvariant="normal">nil</mi><mo stretchy="false">)</mo><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><msub><mi>c</mi> <mi mathvariant="normal">cons</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>a</mi><mo>:</mo><mi>A</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></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 mathvariant="normal">cons</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mi>Γ</mi><mo>⊢</mo><mi>b</mi><mo>:</mo><mi>A</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>g</mi><mo>:</mo><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable><mrow><mi>Γ</mi><mo>⊢</mo><msubsup><mi>β</mi> <mrow><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow> <mi mathvariant="normal">cons</mi></msubsup><mo stretchy="false">(</mo><msub><mi>c</mi> <mi mathvariant="normal">nil</mi></msub><mo>,</mo><msub><mi>c</mi> <mi mathvariant="normal">cons</mi></msub><mo>,</mo><mi>b</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>:</mo><msub><mi mathvariant="normal">Id</mi> <mrow><mi>C</mi><mo stretchy="false">(</mo><mi mathvariant="normal">cons</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msubsup><mi mathvariant="normal">ind</mi> <mrow><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow> <mi>C</mi></msubsup><mo stretchy="false">(</mo><msub><mi>c</mi> <mi mathvariant="normal">nil</mi></msub><mo>,</mo><msub><mi>c</mi> <mi mathvariant="normal">cons</mi></msub><mo>,</mo><mi mathvariant="normal">cons</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>c</mi> <mi mathvariant="normal">cons</mi></msub><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msubsup><mi mathvariant="normal">ind</mi> <mrow><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow> <mi>C</mi></msubsup><mo stretchy="false">(</mo><msub><mi>c</mi> <mi mathvariant="normal">nil</mi></msub><mo>,</mo><msub><mi>c</mi> <mi mathvariant="normal">cons</mi></msub><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{ \begin{array}{l} \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:\mathrm{List}(A) \vdash C(x) \; \mathrm{type} \\ \Gamma \vdash c_\mathrm{nil}:C(\mathrm{nil}) \quad \Gamma \vdash c_\mathrm{cons}:\prod_{a:A} \prod_{x:\mathrm{List}(A))} C(x) \to C(\mathrm{cons}(a)(x)) \\ \Gamma \vdash b:A \quad \Gamma \vdash g:\mathrm{List}(A)) \end{array} }{\Gamma \vdash \beta_{\mathrm{List}(A)}^\mathrm{cons}(c_\mathrm{nil}, c_\mathrm{cons}, b, g):\mathrm{Id}_{C(\mathrm{cons}(b)(g))}(\mathrm{ind}_{\mathrm{List}(A)}^C(c_\mathrm{nil}, c_\mathrm{cons}, \mathrm{cons}(b)(g)), c_\mathrm{cons}(b)(g)(\mathrm{ind}_{\mathrm{List}(A)}^C(c_\mathrm{nil}, c_\mathrm{cons}, g)))} </annotation></semantics></math></div> <p><br /></p> <p>Uniqueness rules for the type of lists:</p> <ul> <li>Judgmental uniqueness rules</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><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 mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></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>g</mi><mo>:</mo><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><mrow><mi>Γ</mi><mo>⊢</mo><msubsup><mi mathvariant="normal">ind</mi> <mrow><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow> <mi>C</mi></msubsup><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">(</mo><mi mathvariant="normal">nil</mi><mo stretchy="false">)</mo><mo>,</mo><mi>λ</mi><mi>a</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>.</mo><mi>c</mi><mo stretchy="false">(</mo><mi mathvariant="normal">cons</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>c</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:\mathrm{List}(A) \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c:\prod_{x:\mathrm{List}(A)} C(x) \quad \Gamma \vdash g:\mathrm{List}(A)}{\Gamma \vdash \mathrm{ind}_{\mathrm{List}(A)}^C(c(\mathrm{nil}), \lambda a:A.\lambda x:\mathrm{List}(A).c(\mathrm{cons}(a)(x)), g) \equiv c(g):C(g)}</annotation></semantics></math></div> <ul> <li>Typal uniqueness rules</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><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 mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></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>g</mi><mo>:</mo><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><mrow><mi>Γ</mi><mo>⊢</mo><msub><mi>η</mi> <mrow><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></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>g</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msubsup><mi mathvariant="normal">ind</mi> <mrow><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow> <mi>C</mi></msubsup><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">(</mo><mi mathvariant="normal">nil</mi><mo stretchy="false">)</mo><mo>,</mo><mi>λ</mi><mi>a</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>λ</mi><mi>x</mi><mo>:</mo><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>.</mo><mi>c</mi><mo stretchy="false">(</mo><mi mathvariant="normal">cons</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>,</mo><mi>c</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:\mathrm{List}(A) \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c:\prod_{x:\mathrm{List}(A)} C(x) \quad \Gamma \vdash g:\mathrm{List}(A)}{\Gamma \vdash \eta_{\mathrm{List}(A)}(c, n):\mathrm{Id}_{C(g)}(\mathrm{ind}_{\mathrm{List}(A)}^C(c(\mathrm{nil}), \lambda a:A.\lambda x:\mathrm{List}(A).c(\mathrm{cons}(a)(x)), g), c(g))}</annotation></semantics></math></div> <p>The elimination, typal computation, and typal uniqueness rules for the type of lists type state that the type of lists satisfy the <strong>dependent universal property of the type of lists</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 type of lists 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>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><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> <mi mathvariant="normal">nil</mi></msub><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mi mathvariant="normal">nil</mi><mo stretchy="false">)</mo><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><msub><mi>c</mi> <mi mathvariant="normal">cons</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>a</mi><mo>:</mo><mi>A</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></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 mathvariant="normal">cons</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><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>ℤ</mi> <mi>C</mi></msubsup><mo stretchy="false">(</mo><msub><mi>c</mi> <mi mathvariant="normal">nil</mi></msub><mo>,</mo><msub><mi>c</mi> <mi mathvariant="normal">cons</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 mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></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><mi mathvariant="normal">nil</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">(</mo><mi mathvariant="normal">nil</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>c</mi> <mi mathvariant="normal">nil</mi></msub><mo stretchy="false">)</mo><mo>×</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>a</mi><mo>:</mo><mi>A</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></munder><msub><mi mathvariant="normal">Id</mi> <mrow><mi>C</mi><mo stretchy="false">(</mo><mi mathvariant="normal">cons</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><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 mathvariant="normal">cons</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>c</mi> <mi mathvariant="normal">cons</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><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 \vdash A \; \mathrm{type} \quad \Gamma, x:\mathrm{List}(A) \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_\mathrm{nil}:C(\mathrm{nil}) \quad \Gamma \vdash c_\mathrm{cons}:\prod_{a:A} \prod_{x:\mathrm{List}(A))} C(x) \to C(\mathrm{cons}(a)(x))}{\Gamma \vdash \mathrm{up}_\mathbb{Z}^C(c_\mathrm{nil}, c_\mathrm{cons}):\exists!c:\prod_{x:\mathrm{List}(A))} C(x).\mathrm{Id}_{C(\mathrm{nil})}(c(\mathrm{nil}), c_\mathrm{nil}) \times \prod_{a:A} \prod_{x:\mathrm{List}(A)} \mathrm{Id}_{C(\mathrm{cons}(a)(x))}(c(\mathrm{cons}(a)(x)), c_\mathrm{cons}(a)(c(x)))}</annotation></semantics></math></div> <h2 id="see_also">See also</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+numbers+type">natural numbers type</a> (the list on the unit type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>𝟙</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{List}(\mathbb{1})</annotation></semantics></math>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/underlying+type+of+free+infinity-group">underlying type of free infinity-group</a> (the invertible version of the type of lists)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/James+construction+type">James construction type</a>, <a class="existingWikiWord" href="/nlab/show/free+monoid">free monoid</a></p> </li> </ul> <h2 id="references">References</h2> <p>Types of lists are defined in section 5.1 in:</p> <ul> <li><em>Homotopy Type Theory: Univalent Foundations of Mathematics</em>, The <a class="existingWikiWord" href="/nlab/show/Univalent+Foundations+Project">Univalent Foundations Project</a>, Institute for Advanced Study, 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>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 25, 2023 at 04:24:46. 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