{"id":7477,"date":"2020-10-26T13:18:23","date_gmt":"2020-10-26T12:18:23","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/?p=7477"},"modified":"2020-12-21T13:19:18","modified_gmt":"2020-12-21T12:19:18","slug":"formatus-pruebas-en-lean-de-la-conmutatividad-de-la-interseccion","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/formatus-pruebas-en-lean-de-la-conmutatividad-de-la-interseccion\/","title":{"rendered":"ForMatUS: Pruebas en Lean de la conmutatividad de la intersecci\u00f3n"},"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\/b9RzBNk6Tzg\">v\u00eddeo<\/a> en el que se comentan pruebas en Lean de la propiedad conmutativa de la intersecci\u00f3n<\/p>\n<pre lang=\"lean\">A \u2229 B = B \u2229 A\n<\/pre>\n<p>usando los estilos declarativos, aplicativos, funcional y autom\u00e1tico.<\/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\/b9RzBNk6Tzg\" 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\/Pruebas_de_la_conmutatividad_de_la_interseccion.lean\">c\u00f3digo<\/a> de la teor\u00eda utilizada<\/p>\n<pre lang=\"lean\">-- ----------------------------------------------------\n-- Ej. 1. Demostrar\n--    A \u2229 B \u2286 B \u2229 A\n-- ----------------------------------------------------\n\nimport data.set\n\nvariable  {U : Type}\nvariables A B : set U\nvariable  x : U\n\nopen set\n\n-- 1\u00aa demostraci\u00f3n\nexample : A \u2229 B \u2286 B \u2229 A :=\nbegin\n  intros x h,\n  simp at *,\n  split,\n  { exact h.right, },\n  { exact h.left,  },\nend\n\n-- 2\u00aa demostraci\u00f3n\nexample : A \u2229 B \u2286 B \u2229 A :=\nbegin\n  intros x h,\n  split,\n  { exact h.right, },\n  { exact h.left,  },\nend\n\n-- 3\u00aa demostraci\u00f3n\nexample : A \u2229 B \u2286 B \u2229 A :=\nbegin\n  rintros x \u27e8h1, h2\u27e9,\n  split,\n  { exact h2, },\n  { exact h1, },\nend\n\n-- 4\u00aa demostraci\u00f3n\nexample : A \u2229 B \u2286 B \u2229 A :=\nbegin\n  rintros x \u27e8h1, h2\u27e9,\n  exact \u27e8h2, h1\u27e9,\nend\n\n-- 5\u00aa demostraci\u00f3n\nexample : A \u2229 B \u2286 B \u2229 A :=\nassume x,\nassume h : x \u2208 A \u2229 B,\nhave h1 : x \u2208 A, from and.left h,\nhave h2 : x \u2208 B, from and.right h,\nshow x \u2208 B \u2229 A,  from and.intro h2 h1\n\n-- 6\u00aa demostraci\u00f3n\nexample : A \u2229 B \u2286 B \u2229 A :=\nassume x,\nassume h : x \u2208 A \u2229 B,\nhave h1 : x \u2208 A \u2227 x \u2208 B, from h,\nhave h2 : x \u2208 B \u2227 x \u2208 A, from and.comm.mp h1,\nshow x \u2208 B \u2229 A,          from h2\n\n-- 7\u00aa demostraci\u00f3n\nexample : A \u2229 B \u2286 B \u2229 A :=\nassume x,\nassume h : x \u2208 A \u2229 B,\nshow x \u2208 B \u2229 A, from and.comm.mp h\n\n-- 8\u00aa demostraci\u00f3n\nexample : A \u2229 B \u2286 B \u2229 A :=\nassume x,\nassume h : x \u2208 A \u2229 B,\nand.comm.mp h\n\n-- 9\u00aa demostraci\u00f3n\nexample : A \u2229 B \u2286 B \u2229 A :=\nassume x,\n\u03bb h, and.comm.mp h\n\n-- 10\u00aa demostraci\u00f3n\nexample : A \u2229 B \u2286 B \u2229 A :=\nassume x,\nand.comm.mp\n\n-- 10\u00aa demostraci\u00f3n\nexample : A \u2229 B \u2286 B \u2229 A :=\n\u03bb _, and.comm.mp\n\n-- 11\u00aa demostraci\u00f3n\nexample : A \u2229 B \u2286 B \u2229 A :=\n-- by hint\nby finish\n\n-- 12\u00aa demostraci\u00f3n\nlemma aux : A \u2229 B \u2286 B \u2229 A :=\nby simp\n\n-- ----------------------------------------------------\n-- Ej. 2. Demostrar\n--    A \u2229 B = B \u2229 A\n-- ----------------------------------------------------\n\n-- 1\u00aa demostraci\u00f3n\nexample : A \u2229 B = B \u2229 A :=\nbegin\n  apply eq_of_subset_of_subset,\n  { exact aux A B, },\n  { exact aux B A, },\nend\n\n-- 2\u00aa demostraci\u00f3n\nexample : A \u2229 B = B \u2229 A :=\neq_of_subset_of_subset (aux A B) (aux B A)\n\n-- 3\u00aa demostraci\u00f3n\nexample : A \u2229 B = B \u2229 A :=\n-- by library_search\ninter_comm A B\n\n-- 4\u00aa demostraci\u00f3n\nexample : A \u2229 B = B \u2229 A :=\n-- by hint\nby finish\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 la propiedad conmutativa de la intersecci\u00f3n A \u2229 B = B \u2229 A usando los estilos declarativos, aplicativos, funcional y autom\u00e1tico. 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;-&#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\/7477"}],"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=7477"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7477\/revisions"}],"predecessor-version":[{"id":7478,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7477\/revisions\/7478"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/media?parent=7477"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/categories?post=7477"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/tags?post=7477"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}