{"id":904,"date":"2022-04-21T10:29:29","date_gmt":"2022-04-21T08:29:29","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=904"},"modified":"2022-04-24T11:12:48","modified_gmt":"2022-04-24T09:12:48","slug":"propiedad-semidistributiva-de-la-interseccion-sobre-la-union","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/propiedad-semidistributiva-de-la-interseccion-sobre-la-union\/","title":{"rendered":"Propiedad semidistributiva de la intersecci\u00f3n sobre la uni\u00f3n"},"content":{"rendered":"<p>Demostrar que <code>s \u2229 (t \u222a u) \u2286 (s \u2229 t) \u222a (s \u2229 u)<\/code>.<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport data.set.basic\nopen set\n\nvariable {\u03b1 : Type}\nvariables s t u : set \u03b1\n\nexample :\n  s \u2229 (t \u222a u) \u2286 (s \u2229 t) \u222a (s \u2229 u) :=\nsorry\n<\/pre>\n<p><!--more--><\/p>\n<h2>1. Soluciones con Lean<\/h2>\n<pre lang=\"lean\">\nimport data.set.basic\nimport tactic\n\nopen set\n\nvariable {\u03b1 : Type}\nvariables s t u : set \u03b1\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample :\n  s \u2229 (t \u222a u) \u2286 (s \u2229 t) \u222a (s \u2229 u) :=\nbegin\n  intros x hx,\n  cases hx with hxs hxtu,\n  cases hxtu with hxt hxu,\n  { left,\n    split,\n    { exact hxs, },\n    { exact hxt, }},\n  { right,\n    split,\n    { exact hxs, },\n    { exact hxu, }},\nend\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample :\n  s \u2229 (t \u222a u) \u2286 (s \u2229 t) \u222a (s \u2229 u) :=\nbegin\n  rintros x \u27e8hxs, hxt | hxu\u27e9,\n  { left,\n    exact \u27e8hxs, hxt\u27e9, },\n  { right,\n    exact \u27e8hxs, hxu\u27e9, },\nend\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample :\n  s \u2229 (t \u222a u) \u2286 (s \u2229 t) \u222a (s \u2229 u) :=\nbegin\n  rintros x \u27e8hxs, hxt | hxu\u27e9,\n  { exact or.inl \u27e8hxs, hxt\u27e9, },\n  { exact or.inr \u27e8hxs, hxu\u27e9, },\nend\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample :\n  s \u2229 (t \u222a u) \u2286 (s \u2229 t) \u222a (s \u2229 u) :=\nbegin\n  intros x hx,\n  by finish,\nend\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample :\n  s \u2229 (t \u222a u) \u2286 (s \u2229 t) \u222a (s \u2229 u) :=\nby rw inter_union_distrib_left\n<\/pre>\n<p>El c\u00f3digo de las demostraciones se encuentra en <a href=\"https:\/\/github.com\/jaalonso\/Demostrando-con-Lean\/blob\/main\/src\/Propiedad_semidistributiva_de_la_interseccion_sobre_la_union.lean\">GitHub<\/a> y puede ejecutarse con el <a href=\"https:\/\/leanprover-community.github.io\/lean-web-editor\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Demostrando-con-Lean\/main\/src\/Propiedad_semidistributiva_de_la_interseccion_sobre_la_union.lean\">Lean Web editor<\/a>.<\/p>\n<p>La construcci\u00f3n de las demostraciones se muestra en el siguiente v\u00eddeo<\/p>\n<p><iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/DRKAjEeeM_8\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<h2>2. Soluciones con Isabelle\/HOL<\/h2>\n<pre lang=\"isar\">\ntheory Propiedad_semidistributiva_de_la_interseccion_sobre_la_union\nimports Main\nbegin\n\n(* 1\u00aa demostraci\u00f3n *)\nlemma \"s \u2229 (t \u222a u) \u2286 (s \u2229 t) \u222a (s \u2229 u)\"\nproof (rule subsetI)\n  fix x\n  assume hx : \"x \u2208 s \u2229 (t \u222a u)\"\n  then have xs : \"x \u2208 s\"\n    by (simp only: IntD1)\n  have xtu: \"x \u2208 t \u222a u\"\n    using hx by (simp only: IntD2)\n  then have \"x \u2208 t \u2228 x \u2208 u\"\n    by (simp only: Un_iff)\n  then show \" x \u2208 s \u2229 t \u222a s \u2229 u\"\n  proof (rule disjE)\n    assume xt : \"x \u2208 t\"\n    have xst : \"x \u2208 s \u2229 t\"\n      using xs xt by (simp only: Int_iff)\n    then show \"x \u2208 (s \u2229 t) \u222a (s \u2229 u)\"\n      by (simp only: UnI1)\n  next\n    assume xu : \"x \u2208 u\"\n    have xst : \"x \u2208 s \u2229 u\"\n      using xs xu by (simp only: Int_iff)\n    then show \"x \u2208 (s \u2229 t) \u222a (s \u2229 u)\"\n      by (simp only: UnI2)\n  qed\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\nlemma \"s \u2229 (t \u222a u) \u2286 (s \u2229 t) \u222a (s \u2229 u)\"\nproof\n  fix x\n  assume hx : \"x \u2208 s \u2229 (t \u222a u)\"\n  then have xs : \"x \u2208 s\"\n    by simp\n  have xtu: \"x \u2208 t \u222a u\"\n    using hx by simp\n  then have \"x \u2208 t \u2228 x \u2208 u\"\n    by simp\n  then show \" x \u2208 s \u2229 t \u222a s \u2229 u\"\n  proof\n    assume xt : \"x \u2208 t\"\n    have xst : \"x \u2208 s \u2229 t\"\n      using xs xt by simp\n    then show \"x \u2208 (s \u2229 t) \u222a (s \u2229 u)\"\n      by simp\n  next\n    assume xu : \"x \u2208 u\"\n    have xst : \"x \u2208 s \u2229 u\"\n      using xs xu by simp\n    then show \"x \u2208 (s \u2229 t) \u222a (s \u2229 u)\"\n      by simp\n  qed\nqed\n\n(* 3\u00aa demostraci\u00f3n *)\nlemma \"s \u2229 (t \u222a u) \u2286 (s \u2229 t) \u222a (s \u2229 u)\"\nproof (rule subsetI)\n  fix x\n  assume hx : \"x \u2208 s \u2229 (t \u222a u)\"\n  then have xs : \"x \u2208 s\"\n    by (simp only: IntD1)\n  have xtu: \"x \u2208 t \u222a u\"\n    using hx by (simp only: IntD2)\n  then show \" x \u2208 s \u2229 t \u222a s \u2229 u\"\n  proof (rule UnE)\n    assume xt : \"x \u2208 t\"\n    have xst : \"x \u2208 s \u2229 t\"\n      using xs xt by (simp only: Int_iff)\n    then show \"x \u2208 (s \u2229 t) \u222a (s \u2229 u)\"\n      by (simp only: UnI1)\n  next\n    assume xu : \"x \u2208 u\"\n    have xst : \"x \u2208 s \u2229 u\"\n      using xs xu by (simp only: Int_iff)\n    then show \"x \u2208 (s \u2229 t) \u222a (s \u2229 u)\"\n      by (simp only: UnI2)\n  qed\nqed\n\n(* 4\u00aa demostraci\u00f3n *)\nlemma \"s \u2229 (t \u222a u) \u2286 (s \u2229 t) \u222a (s \u2229 u)\"\nproof\n  fix x\n  assume hx : \"x \u2208 s \u2229 (t \u222a u)\"\n  then have xs : \"x \u2208 s\"\n    by simp\n  have xtu: \"x \u2208 t \u222a u\"\n    using hx by simp\n  then show \" x \u2208 s \u2229 t \u222a s \u2229 u\"\n  proof (rule UnE)\n    assume xt : \"x \u2208 t\"\n    have xst : \"x \u2208 s \u2229 t\"\n      using xs xt by simp\n    then show \"x \u2208 (s \u2229 t) \u222a (s \u2229 u)\"\n      by simp\n  next\n    assume xu : \"x \u2208 u\"\n    have xst : \"x \u2208 s \u2229 u\"\n      using xs xu by simp\n    then show \"x \u2208 (s \u2229 t) \u222a (s \u2229 u)\"\n      by simp\n  qed\nqed\n\n(* 5\u00aa demostraci\u00f3n *)\nlemma \"s \u2229 (t \u222a u) \u2286 (s \u2229 t) \u222a (s \u2229 u)\"\nby (simp only: Int_Un_distrib)\n\n(* 6\u00aa demostraci\u00f3n *)\nlemma \"s \u2229 (t \u222a u) \u2286 (s \u2229 t) \u222a (s \u2229 u)\"\nby auto\n\nend\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar que s \u2229 (t \u222a u) \u2286 (s \u2229 t) \u222a (s \u2229 u). Para ello, completar la siguiente teor\u00eda de Lean: import data.set.basic open set variable {\u03b1 : Type} variables s t u : set \u03b1 example : s \u2229 (t \u222a u) \u2286 (s \u2229 t) \u222a (s \u2229 u) := sorry<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","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,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[7],"tags":[282],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/904"}],"collection":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/comments?post=904"}],"version-history":[{"count":8,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/904\/revisions"}],"predecessor-version":[{"id":920,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/904\/revisions\/920"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=904"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=904"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=904"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}