{"id":7496,"date":"2020-10-30T14:08:51","date_gmt":"2020-10-30T13:08:51","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/?p=7496"},"modified":"2020-12-21T14:09:33","modified_gmt":"2020-12-21T13:09:33","slug":"formatus-pruebas-en-lean-de-la-interseccion-sobre-union-general-c-%e2%88%a9-%e2%8b%83i-a-i-%e2%8b%83-i-c-%e2%88%a9-a-i","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/formatus-pruebas-en-lean-de-la-interseccion-sobre-union-general-c-%e2%88%a9-%e2%8b%83i-a-i-%e2%8b%83-i-c-%e2%88%a9-a-i\/","title":{"rendered":"ForMatUS: Pruebas en Lean de la intersecci\u00f3n sobre uni\u00f3n general: C \u2229 (\u22c3i, A i) = (\u22c3 i, C \u2229 A i)"},"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\/0jiyLryyubA\">v\u00eddeo<\/a> en el que se comentan pruebas en Lean de la propiedad<\/p>\n<pre lang=\"lean\">C \u2229 (\u22c3i, A i) = (\u22c3 i, C \u2229 A i)\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\/0jiyLryyubA\" 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\/3_Conjuntos\/Pruebas_de_interseccion_sobre_union_general.lean\">c\u00f3digo<\/a> de la teor\u00eda utilizada<\/p>\n<pre lang=\"lean\">import data.set\n\nopen set\n\nvariables {I U : Type}\nvariables {A : I \u2192 set U}\nvariable  {C : set U}\n\n-- ----------------------------------------------------\n-- Ej. 1. Demostrar\n--    C \u2229 (\u22c3i, A i) \u2286 (\u22c3 i, C \u2229 A i)\n-- ----------------------------------------------------\n\n-- 1\u00aa demostraci\u00f3n\nexample :\n  C \u2229 (\u22c3i, A i) \u2286 (\u22c3 i, C \u2229 A i)  :=\nbegin\n  rintros x \u27e8hC, hU\u27e9,\n  rw mem_Union at hU,\n  cases hU with i hA,\n  apply mem_Union.mpr,\n  use i,\n  split,\n  assumption',\nend\n\n-- 2\u00aa demostraci\u00f3n\nexample :\n  C \u2229 (\u22c3i, A i) \u2286 (\u22c3 i, C \u2229 A i)  :=\nbegin\n  intros x h,\n  simp * at *,\nend\n\n-- 3\u00aa demostraci\u00f3n\nlemma inter_Uni_l1 :\n  C \u2229 (\u22c3i, A i) \u2286 (\u22c3 i, C \u2229 A i)  :=\nby {intros x h, simp * at *}\n\n-- ----------------------------------------------------\n-- Ej. 2. Demostrar\n--    (\u22c3 i, C \u2229 A i) \u2286 C \u2229 (\u22c3i, A i)\n-- ----------------------------------------------------\n\n-- 1\u00aa demostraci\u00f3n\nexample :\n  (\u22c3 i, C \u2229 A i) \u2286 C \u2229 (\u22c3i, A i) :=\nbegin\n  intros x h,\n  rw mem_Union at h,\n  cases h with i hi,\n  cases hi with hC hA,\n  split,\n  { exact hC, },\n  { apply mem_Union.mpr,\n    use i,\n    exact hA, },\nend\n\n-- 2\u00aa demostraci\u00f3n\nexample : (\u22c3 i, C \u2229 A i) \u2286 C \u2229 (\u22c3i, A i) :=\nbegin\n  intros x h,\n  rw mem_Union at h,\n  rcases h with \u27e8i, hC, hA\u27e9,\n  split,\n  { exact hC, },\n  { apply mem_Union.mpr,\n    use i,\n    exact hA, },\nend\n\n-- 3\u00aa demostraci\u00f3n\nexample :\n  (\u22c3 i, C \u2229 A i) \u2286 C \u2229 (\u22c3i, A i) :=\nbegin\n  intros x h,\n  simp * at *,\nend\n\n-- 4\u00aa demostraci\u00f3n\nlemma inter_Uni_l2 :\n  (\u22c3 i, C \u2229 A i) \u2286 C \u2229 (\u22c3i, A i) :=\nby {intros x h, simp * at *}\n\n-- ----------------------------------------------------\n-- Ej. 3. Demostrar\n--    C \u2229 (\u22c3i, A i) = (\u22c3 i, C \u2229 A i)\n-- ----------------------------------------------------\n\n-- 1\u00aa demostraci\u00f3n\nexample :\n  C \u2229 (\u22c3i, A i) = (\u22c3 i, C \u2229 A i) :=\neq_of_subset_of_subset inter_Uni_l1 inter_Uni_l2\n\n-- 2\u00aa demostraci\u00f3n\nexample :\n  C \u2229 (\u22c3i, A i) = (\u22c3 i, C \u2229 A i) :=\n-- by library_search\ninter_Union C A\n\n-- 3\u00aa demostraci\u00f3n\nexample :\n  C \u2229 (\u22c3i, A i) = (\u22c3 i, C \u2229 A i) :=\next $ by simp\n\n-- 4\u00aa demostraci\u00f3n\nexample :\n  C \u2229 (\u22c3i, A i) = (\u22c3 i, C \u2229 A i) :=\nby {ext, simp}\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 la propiedad C \u2229 (\u22c3i, A i) = (\u22c3 i, C \u2229 A i) 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&#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\/7496"}],"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=7496"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7496\/revisions"}],"predecessor-version":[{"id":7498,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7496\/revisions\/7498"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/media?parent=7496"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/categories?post=7496"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/tags?post=7496"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}