{"id":7521,"date":"2020-11-02T19:11:14","date_gmt":"2020-11-02T18:11:14","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/?p=7521"},"modified":"2020-12-21T19:11:59","modified_gmt":"2020-12-21T18:11:59","slug":"formatus-pruebas-en-lean-de-que-las-relaciones-reflexivas-y-euclideas-son-de-equivalencia","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/formatus-pruebas-en-lean-de-que-las-relaciones-reflexivas-y-euclideas-son-de-equivalencia\/","title":{"rendered":"ForMatUS: Pruebas en Lean de que las relaciones reflexivas y eucl\u00eddeas son de equivalencia"},"content":{"rendered":"<p>He a\u00f1adido a la lista <a href=\"https:\/\/bit.ly\/2FcUrwQ\">L\u00f3gica con Lean<\/a> el <a href=\"https:\/\/youtu.be\/OxcHaXGmmu4\">v\u00eddeo<\/a> en el que se comentan pruebas en Lean de que si una relaci\u00f3n es reflexiva y eucl\u00eddea, entonces es de equivalencia. Se usan los estilos declarativos, aplicativos y funcional.<\/p>\n<p>A continuaci\u00f3n, se muestra el v\u00eddeo<\/p>\n<p><center><\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/OxcHaXGmmu4\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<p><\/center>y el <a href=\"https:\/\/github.com\/jaalonso\/Logica_con_Lean\/blob\/master\/src\/4_Relaciones\/Las_relaciones_reflexivas_y_euclideas_son_de_equivalencia.lean\">c\u00f3digo<\/a> de la teor\u00eda utilizada<\/p>\n<pre lang=\"lean\">-- ----------------------------------------------------\n-- Una relaci\u00f3n binaria (\u2248) es eucl\u00eddea si\n--    \u2200 {a b c}, a \u2248 b \u2192 c \u2248 b \u2192 a \u2248 c\n--\n-- El objetivo de esta teor\u00eda es demostrar que si una\n-- relaci\u00f3n es reflexiva y eucl\u00eddea, entonces es de\n-- equivalencia.\n-- ----------------------------------------------------\n\nimport tactic\n\nsection\n\nparameter {A : Type}\nparameter (R : A \u2192 A \u2192 Prop)\n\nlocal infix \u2248 := R\n\nparameter reflexivaR : reflexive (\u2248)\nparameter euclideaR : \u2200 {a b c}, a \u2248 b \u2192 c \u2248 b \u2192 a \u2248 c\n\ninclude reflexivaR euclideaR\n\n-- ----------------------------------------------------\n-- Ej. 1. Demostrar que las relaciones reflexivas y\n-- y eucl\u00eddeas son sim\u00e9tricas.\n-- ----------------------------------------------------\n\n-- 1\u00aa demostraci\u00f3n\nexample : symmetric (\u2248) :=\nbegin\n  intros a b h,\n  exact euclideaR (reflexivaR b) h,\nend\n\n-- 2\u00aa demostraci\u00f3n\nexample : symmetric (\u2248) :=\n\u03bb a b h, euclideaR (reflexivaR b) h\n\n-- 3\u00aa demostraci\u00f3n\nlemma simetricaR : symmetric (\u2248) :=\nassume a b (h1 : a \u2248 b),\nhave h2 : b \u2248 b, from (reflexivaR b),\nshow b \u2248 a, from euclideaR h2 h1\n\n-- ----------------------------------------------------\n-- Ej. 2. Demostrar que las relaciones reflexivas y\n-- y eucl\u00eddeas son transitivas.\n-- ----------------------------------------------------\n\n-- 1\u00aa demostraci\u00f3n\nexample : transitive (\u2248) :=\nbegin\n  rintros a b c h1 h2,\n  apply euclideaR h1,\n  exact euclideaR (reflexivaR c) h2,\nend\n\n-- 2\u00aa demostraci\u00f3n\nlemma transitivaR : transitive (\u2248) :=\n\u03bb a b c h1 h2, (euclideaR h1) (euclideaR (reflexivaR c) h2)\n\n-- 3\u00aa demostraci\u00f3n\nexample : transitive (\u2248) :=\nassume a b c (h1 : a \u2248 b) (h2 : b \u2248 c),\nhave h3 : c \u2248 b, from euclideaR (reflexivaR c) h2,\nshow a \u2248 c, from euclideaR h1 h3\n\n-- ----------------------------------------------------\n-- Ej. 3. Demostrar que las relaciones reflexivas y\n-- y eucl\u00eddeas son de equivalencia.\n-- ----------------------------------------------------\n\n-- 1\u00aa demostraci\u00f3n\nexample : equivalence (\u2248) :=\nbegin\n  unfold equivalence,\n  exact \u27e8reflexivaR, simetricaR, transitivaR\u27e9,\nend\n\n-- 2\u00aa demostraci\u00f3n\nexample : equivalence (\u2248) :=\n\u27e8reflexivaR, simetricaR, transitivaR\u27e9\n\nend\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>He a\u00f1adido a la lista L\u00f3gica con Lean el v\u00eddeo en el que se comentan pruebas en Lean de que si una relaci\u00f3n es reflexiva y eucl\u00eddea, entonces es de equivalencia. Se usan los estilos declarativos, aplicativos y funcional. A continuaci\u00f3n, se muestra el v\u00eddeo y el c\u00f3digo de la teor\u00eda utilizada &#8212; &#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;- &#8211;&#8230;<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"footnotes":"","_jetpack_memberships_contains_paid_content":false},"categories":[335],"tags":[166,336],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_likes_enabled":false,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7521"}],"collection":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/comments?post=7521"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7521\/revisions"}],"predecessor-version":[{"id":7522,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7521\/revisions\/7522"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/media?parent=7521"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/categories?post=7521"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/tags?post=7521"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}