{"id":7441,"date":"2020-10-02T17:21:39","date_gmt":"2020-10-02T15:21:39","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/?p=7441"},"modified":"2020-12-20T17:22:25","modified_gmt":"2020-12-20T16:22:25","slug":"formatus-regla-de-eliminacion-del-cuantificador-existencial-en-lean","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/formatus-regla-de-eliminacion-del-cuantificador-existencial-en-lean\/","title":{"rendered":"ForMatUS: Regla de eliminaci\u00f3n del cuantificador existencial 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\/WWWqZ4BJAAM\">v\u00eddeo<\/a> en el que se comentan 11 pruebas en Lean de la propiedad<\/p>\n<pre lang=\"lean\">\u2200x [P(x) \u2192 Q(x)], \u2203x P(x) \u22a2 \u2203x Q(x)\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\/WWWqZ4BJAAM\" 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\/2_LPO\/Regla_de_eliminacion_del_cuantificador_existencial.lean\">c\u00f3digo<\/a> de la teor\u00eda utilizada<\/p>\n<pre lang=\"lean\">-- Ej. 1. Demostrar\n--    \u2200x [P(x) \u2192 Q(x)], \u2203x P(x) \u22a2 \u2203x Q(x)\n\nimport tactic\n\nvariable  U : Type\nvariables P Q : U -&gt; Prop\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h1 : \u2200x, P x \u2192 Q x)\n  (h2 : \u2203x, P x)\n  : \u2203x, Q x :=\nexists.elim h2\n  ( assume x\u2080 (h3 : P x\u2080),\n    have h4 : P x\u2080 \u2192 Q x\u2080, from h1 x\u2080,\n    have h5 : Q x\u2080,        from h4 h3,\n    show \u2203x, Q x,          from exists.intro x\u2080 h5 )\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h1 : \u2200x, P x \u2192 Q x)\n  (h2 : \u2203x, P x)\n  : \u2203x, Q x :=\nexists.elim h2\n  ( assume x\u2080 (h3 : P x\u2080),\n    have h4 : P x\u2080 \u2192 Q x\u2080, from h1 x\u2080,\n    have h5 : Q x\u2080,        from h4 h3,\n    show \u2203x, Q x,          from \u27e8x\u2080, h5\u27e9 )\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h1 : \u2200x, P x \u2192 Q x)\n  (h2 : \u2203x, P x)\n  : \u2203x, Q x :=\nexists.elim h2\n  ( assume x\u2080 (h3 : P x\u2080),\n    have h4 : P x\u2080 \u2192 Q x\u2080, from h1 x\u2080,\n    have h5 : Q x\u2080,        from h4 h3,\n    \u27e8x\u2080, h5\u27e9 )\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h1 : \u2200x, P x \u2192 Q x)\n  (h2 : \u2203x, P x)\n  : \u2203x, Q x :=\nexists.elim h2\n  ( assume x\u2080 (h3 : P x\u2080),\n    have h4 : P x\u2080 \u2192 Q x\u2080, from h1 x\u2080,\n    \u27e8x\u2080, h4 h3\u27e9 )\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h1 : \u2200x, P x \u2192 Q x)\n  (h2 : \u2203x, P x)\n  : \u2203x, Q x :=\nexists.elim h2\n  ( assume x\u2080 (h3 : P x\u2080),\n    \u27e8x\u2080, h1 x\u2080 h3\u27e9 )\n\n-- 6\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h1 : \u2200x, P x \u2192 Q x)\n  (h2 : \u2203x, P x)\n  : \u2203x, Q x :=\nexists.elim h2 (\u03bb x\u2080 h3, \u27e8x\u2080, h1 x\u2080 h3\u27e9)\n\n-- 7\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h1 : \u2200x, P x \u2192 Q x)\n  (h2 : \u2203x, P x)\n  : \u2203x, Q x :=\n-- by library_search\nExists.imp h1 h2\n\n-- 8\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h1 : \u2200x, P x \u2192 Q x)\n  (h2 : \u2203x, P x)\n  : \u2203x, Q x :=\nbegin\n  cases h2 with x\u2080 h3,\n  use x\u2080,\n  apply h1,\n  exact h3,\nend\n\n-- 9\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h1 : \u2200x, P x \u2192 Q x)\n  (h2 : \u2203x, P x)\n  : \u2203x, Q x :=\nbegin\n  cases h2 with x\u2080 h3,\n  use x\u2080,\n  specialize h1 x\u2080,\n  apply h1,\n  exact h3,\nend\n\n-- 10\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h1 : \u2200x, P x \u2192 Q x)\n  (h2 : \u2203x, P x)\n  : \u2203x, Q x :=\n-- by hint\nby tauto\n\n-- 11\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h1 : \u2200x, P x \u2192 Q x)\n  (h2 : \u2203x, P x)\n  : \u2203x, Q x :=\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 11 pruebas en Lean de la propiedad \u2200x [P(x) \u2192 Q(x)], \u2203x P(x) \u22a2 \u2203x Q(x) 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; Ej. 1&#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":[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\/7441"}],"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=7441"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7441\/revisions"}],"predecessor-version":[{"id":7442,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7441\/revisions\/7442"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/media?parent=7441"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/categories?post=7441"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/tags?post=7441"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}