{"id":922,"date":"2022-04-26T08:50:46","date_gmt":"2022-04-26T06:50:46","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=922"},"modified":"2022-04-26T10:30:56","modified_gmt":"2022-04-26T08:30:56","slug":"interseccion-con-su-union","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/interseccion-con-su-union\/","title":{"rendered":"Intersecci\u00f3n con su uni\u00f3n"},"content":{"rendered":"<p>Demostrar que <code>s \u2229 (s \u222a t) = s<\/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 : set \u03b1\n\nexample : s \u2229 (s \u222a t) = s :=\nsorry\n<\/pre>\n<p><!--more--><\/p>\n<p><strong>Soluciones con Lean<\/strong><\/p>\n<pre lang=\"lean\">\nimport data.set.basic\nimport tactic\nopen set\n\nvariable {\u03b1 : Type}\nvariables s t : set \u03b1\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (s \u222a t) = s :=\nbegin\n  ext x,\n  split,\n  { intros h,\n    dsimp at h,\n    exact h.1, },\n  { intro xs,\n    dsimp,\n    split,\n    { exact xs, },\n    { left,\n      exact xs, }},\nend\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (s \u222a t) = s :=\nbegin\n  ext x,\n  split,\n  { intros h,\n    exact h.1, },\n  { intro xs,\n    split,\n    { exact xs, },\n    { left,\n      exact xs, }},\nend\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (s \u222a t) = s :=\nbegin\n  ext x,\n  split,\n  { intros h,\n    exact h.1, },\n  { intro xs,\n    split,\n    { exact xs, },\n    { exact (or.inl xs), }},\nend\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (s \u222a t) = s :=\nbegin\n  ext,\n  exact \u27e8\u03bb h, h.1,\n         \u03bb xs, \u27e8xs, or.inl xs\u27e9\u27e9,\nend\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (s \u222a t) = s :=\nbegin\n  ext,\n  exact \u27e8and.left,\n         \u03bb xs, \u27e8xs, or.inl xs\u27e9\u27e9,\nend\n\n-- 6\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (s \u222a t) = s :=\nbegin\n  ext x,\n  split,\n  { rintros \u27e8xs, _\u27e9,\n    exact xs },\n  { intro xs,\n    use xs,\n    left,\n    exact xs },\nend\n\n-- 7\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (s \u222a t) = s :=\nbegin\n  apply subset_antisymm,\n  { rintros x \u27e8hxs,-\u27e9,\n    exact hxs, },\n  { intros x hxs,\n    exact \u27e8hxs, or.inl hxs\u27e9, },\nend\n\n-- 8\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (s \u222a t) = s :=\n-- by suggest\ninf_sup_self\n\n-- 9\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (s \u222a t) = s :=\n-- by hint\nby finish\n<\/pre>\n<p>El c\u00f3digo de las demostraciones se encuentra en <a href=\"https:\/\/github.com\/jaalonso\/Razonando-con-Lean\/blob\/main\/src\/Interseccion_con_su_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\/Razonando-con-Lean\/main\/src\/Interseccion_con_su_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\/pQ4z5NCE5fU\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<p><strong>Soluciones con Isabelle\/HOL<\/strong><\/p>\n<pre lang=\"isar\">\ntheory Interseccion_con_su_union\nimports Main\nbegin\n\n(* 1\u00aa demostraci\u00f3n *)\nlemma \"s \u2229 (s \u222a t) = s\"\nproof (rule  equalityI)\n  show \"s \u2229 (s \u222a t) \u2286 s\"\n  proof (rule subsetI)\n    fix x\n    assume \"x \u2208 s \u2229 (s \u222a t)\"\n    then show \"x \u2208 s\"\n      by (simp only: IntD1)\n  qed\nnext\n  show \"s \u2286 s \u2229 (s \u222a t)\"\n  proof (rule subsetI)\n    fix x\n    assume \"x \u2208 s\"\n    then have \"x \u2208 s \u222a t\"\n      by (simp only: UnI1)\n    with \u2039x \u2208 s\u203a show \"x \u2208 s \u2229 (s \u222a t)\"\n      by (rule IntI)\n  qed\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\nlemma \"s \u2229 (s \u222a t) = s\"\nproof\n  show \"s \u2229 (s \u222a t) \u2286 s\"\n  proof\n    fix x\n    assume \"x \u2208 s \u2229 (s \u222a t)\"\n    then show \"x \u2208 s\"\n      by simp\n  qed\nnext\n  show \"s \u2286 s \u2229 (s \u222a t)\"\n  proof\n    fix x\n    assume \"x \u2208 s\"\n    then have \"x \u2208 s \u222a t\"\n      by simp\n    then show \"x \u2208 s \u2229 (s \u222a t)\"\n      using \u2039x \u2208 s\u203a by simp\n  qed\nqed\n\n(* 3\u00aa demostraci\u00f3n *)\nlemma \"s \u2229 (s \u222a t) = s\"\nby (fact Un_Int_eq)\n\n(* 4\u00aa demostraci\u00f3n *)\nlemma \"s \u2229 (s \u222a t) = s\"\nby auto\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar que s \u2229 (s \u222a t) = s Para ello, completar la siguiente teor\u00eda de Lean: import data.set.basic open set variable {\u03b1 : Type} variables s t : set \u03b1 example : s \u2229 (s \u222a t) = s := 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\/922"}],"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=922"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/922\/revisions"}],"predecessor-version":[{"id":925,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/922\/revisions\/925"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=922"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=922"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=922"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}