{"id":7525,"date":"2020-11-03T08:28:37","date_gmt":"2020-11-03T07:28:37","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/?p=7525"},"modified":"2020-12-22T08:29:24","modified_gmt":"2020-12-22T07:29:24","slug":"formatus-pruebas-en-lean-de-que-la-identidad-es-biyectiva","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/formatus-pruebas-en-lean-de-que-la-identidad-es-biyectiva\/","title":{"rendered":"ForMatUS: Pruebas en Lean de que la identidad es biyectiva"},"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\/KrjzxBs_EuU\">v\u00eddeo<\/a> en el que se comentan pruebas en Lean de que la identidad es biyectiva 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\/KrjzxBs_EuU\" 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\/5_Funciones\/La_identidad_es_biyectiva.lean\">c\u00f3digo<\/a> de la teor\u00eda utilizada<\/p>\n<pre lang=\"lean\">import tactic\n\nopen function\n\nvariables {X : Type}\n\n-- #print injective\n-- #print surjective\n-- #print bijective\n\n-- ----------------------------------------------------\n-- Ej. 1. Demostrar que la identidad es inyectiva.\n-- ----------------------------------------------------\n\n-- 1\u00aa demostraci\u00f3n\nexample : injective (@id X) :=\nbegin\n  intros x\u2081 x\u2082 h,\n  exact h,\nend\n\n-- 2\u00aa demostraci\u00f3n\nexample : injective (@id X) :=\n\u03bb x\u2081 x\u2082 h, h\n\n-- 3\u00aa demostraci\u00f3n\nexample : injective (@id X) :=\n\u03bb x\u2081 x\u2082, id\n\n-- 4\u00aa demostraci\u00f3n\nexample : injective (@id X) :=\nassume x\u2081 x\u2082,\nassume h : id x\u2081 = id x\u2082,\nshow x\u2081 = x\u2082, from h\n\n-- 5\u00aa demostraci\u00f3n\nexample : injective (@id X) :=\n--by library_search\ninjective_id\n\n-- 6\u00aa demostraci\u00f3n\nexample : injective (@id X) :=\n-- by hint\nby tauto\n\n-- 7\u00aa demostraci\u00f3n\nexample : injective (@id X) :=\n-- by hint\nby finish\n\n-- ----------------------------------------------------\n-- Ej. 2. Demostrar que la identidad es suprayectiva.\n-- ----------------------------------------------------\n\n-- 1\u00aa demostraci\u00f3n\nexample : surjective (@id X) :=\nbegin\n  intro x,\n  use x,\n  exact rfl,\nend\n\n-- 2\u00aa demostraci\u00f3n\nexample : surjective (@id X) :=\nbegin\n  intro x,\n  exact \u27e8x, rfl\u27e9,\nend\n\n-- 3\u00aa demostraci\u00f3n\nexample : surjective (@id X) :=\n\u03bb x, \u27e8x, rfl\u27e9\n\n-- 4\u00aa demostraci\u00f3n\nexample : surjective (@id X) :=\nassume y,\nshow \u2203 x, id x = y, from exists.intro y rfl\n\n-- 5\u00aa demostraci\u00f3n\nexample : surjective (@id X) :=\n-- by library_search\nsurjective_id\n\n-- 6\u00aa demostraci\u00f3n\nexample : surjective (@id X) :=\n-- by hint\nby tauto\n\n-- ----------------------------------------------------\n-- Ej. 3. Demostrar que la identidad es biyectiva.\n-- ----------------------------------------------------\n\n-- 1\u00aa demostraci\u00f3n\nexample : bijective (@id X) :=\nand.intro injective_id surjective_id\n\n-- 2\u00aa demostraci\u00f3n\nexample : bijective (@id X) :=\n\u27e8injective_id, surjective_id\u27e9\n\n-- 3\u00aa demostraci\u00f3n\nexample : bijective (@id X) :=\n-- by library_search\nbijective_id\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 la identidad es biyectiva 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 tactic open function variables {X : Type} &#8212; #print&#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\/7525"}],"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=7525"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7525\/revisions"}],"predecessor-version":[{"id":7526,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7525\/revisions\/7526"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/media?parent=7525"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/categories?post=7525"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/tags?post=7525"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}