{"id":7489,"date":"2020-10-29T13:48:23","date_gmt":"2020-10-29T12:48:23","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/?p=7489"},"modified":"2020-12-21T13:49:12","modified_gmt":"2020-12-21T12:49:12","slug":"formatus-union-e-interseccion-de-familias-de-conjuntos-en-lean","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/formatus-union-e-interseccion-de-familias-de-conjuntos-en-lean\/","title":{"rendered":"ForMatUS: Uni\u00f3n e intersecci\u00f3n de familias de conjuntos en Lean"},"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\/AnB6Gm477M4\">v\u00eddeo<\/a> en el que se comentan c\u00f3mo definir en Lean la uni\u00f3n e intersecci\u00f3n de familias de conjuntos y caracterizar la relaci\u00f3n de pertenencia.<\/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\/AnB6Gm477M4\" 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\/3_Conjuntos\/Union_e_interseccion_de_familias_de_conjuntos.lean\">c\u00f3digo<\/a> de la teor\u00eda utilizada<\/p>\n<pre lang=\"lean\">-- ----------------------------------------------------\n-- Ej. 1. Declarar I y U como variables de tipo.\n-- ----------------------------------------------------\n\nvariables {I U : Type}\n\n-- ----------------------------------------------------\n-- Ej. 2. Definir la funci\u00f3n\n--    Union : (I \u2192 set U) \u2192 set U\n-- tal que (Union A) es la uni\u00f3n de de los conjuntos de\n-- la familia A.\n-- ----------------------------------------------------\n\ndef Union (A : I \u2192 set U) : set U :=\n  { x | \u2203 i : I, x \u2208 A i }\n\n-- ----------------------------------------------------\n-- Ej. 3. Definir la funci\u00f3n\n--    Inter : (I \u2192 set U) \u2192 set U\n-- tal que (Inter A) es la intersecci\u00f3n de de los\n-- conjuntos de la familia A.\n-- ----------------------------------------------------\n\ndef Inter (A : I \u2192 set U) : set U :=\n  { x | \u2200 i : I, x \u2208 A i }\n\n-- ----------------------------------------------------\n-- Ej. 4. Declarar\n-- + x como una variable sobre U y\n-- + A como una variable sobre familas de conjuntos de\n--   U con \u00edndice en A.\n-- ----------------------------------------------------\n\nvariable x : U\nvariable (A : I \u2192 set U)\n\n-- ----------------------------------------------------\n-- Ej. 5. Demostrar que\n--    x \u2208 Union A \u22a2 \u2203 i, x \u2208 A i\n-- ----------------------------------------------------\n\nexample\n  (h : x \u2208 Union A)\n  : \u2203 i, x \u2208 A i :=\nh\n\n-- ----------------------------------------------------\n-- Ej. 6. Demostrar que\n--    x \u2208 x \u2208 Inter A \u22a2 \u2200 i, x \u2208 A i\n-- ----------------------------------------------------\n\nexample\n  (h : x \u2208 Inter A)\n  : \u2200 i, x \u2208 A i :=\nh\n\n-- ----------------------------------------------------\n-- Ej 7. Usar (\u22c3 i, A i) como notaci\u00f3n para (Union A).\n-- ----------------------------------------------------\n\nnotation `\u22c3` binders `, ` r:(scoped f, Union f) := r\n\n-- ----------------------------------------------------\n-- Ej 8. Usar (\u22c2 i, A i) como notaci\u00f3n para (Inter A).\n-- ----------------------------------------------------\n\nnotation `\u22c2` binders `, ` r:(scoped f, Inter f) := r\n\n\n-- ----------------------------------------------------\n-- Ej. 9. Demostrar que\n--    x \u2208 \u22c3 i, A i \u22a2 \u2203 i, x \u2208 A i\n-- ----------------------------------------------------\n\nexample\n  (h : x \u2208 \u22c3 i, A i)\n  : \u2203 i, x \u2208 A i :=\nh\n\n-- ----------------------------------------------------\n-- Ej. 10. Demostrar que\n--    x \u2208 x \u2208 \u22c2 i, A i \u22a2 \u2200 i, x \u2208 A i\n-- ----------------------------------------------------\n\nexample\n  (h : x \u2208 \u22c2 i, A i)\n  : \u2200 i, x \u2208 A i :=\nh\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 c\u00f3mo definir en Lean la uni\u00f3n e intersecci\u00f3n de familias de conjuntos y caracterizar la relaci\u00f3n de pertenencia. 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;- &#8212; Ej. 1. Declarar I y U&#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\/7489"}],"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=7489"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7489\/revisions"}],"predecessor-version":[{"id":7490,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7489\/revisions\/7490"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/media?parent=7489"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/categories?post=7489"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/tags?post=7489"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}