{"id":7502,"date":"2020-10-30T16:25:38","date_gmt":"2020-10-30T15:25:38","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/?p=7502"},"modified":"2020-12-21T16:26:26","modified_gmt":"2020-12-21T15:26:26","slug":"formatus-pruebas-en-lean-de-%f0%9d%92%ab-a-%e2%8a%86-%f0%9d%92%ab-b-%e2%86%94-a-%e2%8a%86-b","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/formatus-pruebas-en-lean-de-%f0%9d%92%ab-a-%e2%8a%86-%f0%9d%92%ab-b-%e2%86%94-a-%e2%8a%86-b\/","title":{"rendered":"ForMatUS: Pruebas en Lean de \ud835\udcab A \u2286 \ud835\udcab B \u2194 A \u2286 B"},"content":{"rendered":"<div id=\"content\">\n<p>He a\u00f1adido a la lista <a href=\"https:\/\/bit.ly\/2FcUrwQ\">L\u00f3gica con Lean<\/a> el <a href=\"https:\/\/youtu.be\/2b8GQdRazxQ\">v\u00eddeo<\/a> en el que se comentan pruebas de la monoton\u00eda del conjunto potencia:<\/p>\n<pre lang=\"lean\">\ud835\udcab A \u2286 \ud835\udcab B \u2194 A \u2286 B\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><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/2b8GQdRazxQ\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>y el <a href=\"https:\/\/github.com\/jaalonso\/Logica_con_Lean\/blob\/master\/src\/3_Conjuntos\/Monotonia_del_conjunto_potencia.lean\">c\u00f3digo<\/a> de la teor\u00eda utilizada<\/p>\n<pre lang=\"lean\">import data.set\nopen set\n\nvariable  {U : Type}\nvariables {A B C : set U}\n\n-- #reduce \ud835\udcab A\n-- #reduce B \u2208 \ud835\udcab A\n\n-- ----------------------------------------------------\n-- Ej. 1. Demostrar\n--    \ud835\udcab A \u2286 \ud835\udcab B \u2192 A \u2286 B\n-- ----------------------------------------------------\n\n-- 1\u00aa demostraci\u00f3n\nexample : \ud835\udcab A \u2286 \ud835\udcab B \u2192 A \u2286 B :=\nbegin\n  intro h,\n  apply subset_of_mem_powerset,\n  apply h,\n  apply mem_powerset,\n  exact subset.rfl,\nend\n\n-- 2\u00aa demostraci\u00f3n\nexample : \ud835\udcab A \u2286 \ud835\udcab B \u2192 A \u2286 B :=\nbegin\n  intro h,\n  apply h,\n  exact subset.rfl,\nend\n\n-- 3\u00aa demostraci\u00f3n\nexample : \ud835\udcab A \u2286 \ud835\udcab B \u2192 A \u2286 B :=\nbegin\n  intro h,\n  exact (h subset.rfl),\nend\n\n-- 4\u00aa demostraci\u00f3n\nexample : \ud835\udcab A \u2286 \ud835\udcab B \u2192 A \u2286 B :=\n\u03bb h, h subset.rfl\n\n-- 5\u00aa demostraci\u00f3n\nexample : \ud835\udcab A \u2286 \ud835\udcab B \u2192 A \u2286 B :=\nassume h1 : \ud835\udcab A \u2286 \ud835\udcab B,\nhave h2 : A \u2286 A, from subset.rfl,\nhave h3 : A \u2208 \ud835\udcab A, from h2,\nhave h4 : A \u2208 \ud835\udcab B, from h1 h3,\nshow A \u2286 B, from h4\n\n-- 6\u00aa demostraci\u00f3n\nexample : \ud835\udcab A \u2286 \ud835\udcab B \u2192 A \u2286 B :=\nassume h1 : \ud835\udcab A \u2286 \ud835\udcab B,\nhave h2 : A \u2286 A, from subset.rfl,\nhave h3 : A \u2208 \ud835\udcab A, from h2,\nh1 h3\n\n-- 7\u00aa demostraci\u00f3n\nexample : \ud835\udcab A \u2286 \ud835\udcab B \u2192 A \u2286 B :=\nassume h1 : \ud835\udcab A \u2286 \ud835\udcab B,\nhave h2 : A \u2286 A, from subset.rfl,\nh1 h2\n\n-- 8\u00aa demostraci\u00f3n\nexample : \ud835\udcab A \u2286 \ud835\udcab B \u2192 A \u2286 B :=\nassume h1 : \ud835\udcab A \u2286 \ud835\udcab B,\nh1 subset.rfl\n\n-- 9\u00aa demostraci\u00f3n\nlemma aux1 : \ud835\udcab A \u2286 \ud835\udcab B \u2192 A \u2286 B :=\n\u03bb h, h subset.rfl\n\n-- 10\u00aa demostraci\u00f3n\nexample : \ud835\udcab A \u2286 \ud835\udcab B \u2192 A \u2286 B :=\npowerset_mono.mp\n\n-- ----------------------------------------------------\n-- Ej. 2. Demostrar\n--    A \u2286 B \u2192 \ud835\udcab A \u2286 \ud835\udcab B\n-- ----------------------------------------------------\n\n-- 1\u00aa demostraci\u00f3n\nexample : A \u2286 B \u2192 \ud835\udcab A \u2286 \ud835\udcab B :=\nbegin\n  intro h,\n  intros C hCA,\n  apply mem_powerset,\n  apply subset.trans hCA h,\nend\n\n-- 2\u00aa demostraci\u00f3n\nexample : A \u2286 B \u2192 \ud835\udcab A \u2286 \ud835\udcab B :=\nbegin\n  intros h C hCA,\n  apply subset.trans hCA h,\nend\n\n-- 3\u00aa demostraci\u00f3n\nlemma aux2 : A \u2286 B \u2192 \ud835\udcab A \u2286 \ud835\udcab B :=\n\u03bb h C hCA, subset.trans hCA h\n\n-- 4\u00aa demostraci\u00f3n\nexample : A \u2286 B \u2192 \ud835\udcab A \u2286 \ud835\udcab B :=\npowerset_mono.mpr\n\n-- ----------------------------------------------------\n-- Ej. 3. Demostrar\n--    \ud835\udcab A \u2286 \ud835\udcab B \u2194 A \u2286 B\n-- ----------------------------------------------------\n\n-- 1\u00aa demostraci\u00f3n\nexample : \ud835\udcab A \u2286 \ud835\udcab B \u2194 A \u2286 B :=\niff.intro aux1 aux2\n\n-- 2\u00aa demostraci\u00f3n\nexample : \ud835\udcab A \u2286 \ud835\udcab B \u2194 A \u2286 B :=\n-- by library_search\npowerset_mono\n\n-- 3\u00aa demostraci\u00f3n\nexample : \ud835\udcab A \u2286 \ud835\udcab B \u2194 A \u2286 B :=\n-- by hint\nby finish\n\n-- 4\u00aa demostraci\u00f3n\nexample : \ud835\udcab A \u2286 \ud835\udcab B \u2194 A \u2286 B :=\nby simp\n<\/pre>\n<\/div>\n<div id=\"postamble\" class=\"status\"><\/div>\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 de la monoton\u00eda del conjunto potencia: \ud835\udcab A \u2286 \ud835\udcab B \u2194 A \u2286 B 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 import data.set open&#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\/7502"}],"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=7502"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7502\/revisions"}],"predecessor-version":[{"id":7503,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7502\/revisions\/7503"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/media?parent=7502"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/categories?post=7502"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/tags?post=7502"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}