{"id":1141,"date":"2022-10-21T06:00:39","date_gmt":"2022-10-21T04:00:39","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1141"},"modified":"2022-10-20T11:43:31","modified_gmt":"2022-10-20T09:43:31","slug":"si-r-es-un-reticulo-y-x-y-%e2%88%88-r-entonces-x-%e2%8a%94-x-%e2%8a%93-y-x","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/si-r-es-un-reticulo-y-x-y-%e2%88%88-r-entonces-x-%e2%8a%94-x-%e2%8a%93-y-x\/","title":{"rendered":"Si R es un ret\u00edculo y x, y \u2208 R, entonces x \u2294 (x \u2293 y) = x"},"content":{"rendered":"<p>Demostrar que si R es un ret\u00edculo y x, y \u2208 R, entonces<\/p>\n<pre lang=\"text\">\n   x \u2294 (x \u2293 y) = x\n<\/pre>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport order.lattice\nvariables {R : Type*} [lattice R]\nvariables x y : R\n\nexample : x \u2294 (x \u2293 y) = x :=\nsorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Soluciones con Lean<\/b><\/p>\n<pre lang=\"lean\">\nimport order.lattice\nvariables {R : Type*} [lattice R]\nvariables x y : R\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : x \u2294 (x \u2293 y) = x :=\nbegin\n  have h1 : x \u2294 (x \u2293 y) \u2264 x, finish,\n  have h2 : x \u2264 x \u2294 (x \u2293 y), finish,\n  show x \u2294 (x \u2293 y) = x,\n    by exact le_antisymm h1 h2,\nend\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : x \u2294 (x \u2293 y) = x :=\nbegin\n  have h1 : x \u2294 (x \u2293 y) \u2264 x,\n  { have h1a : x \u2264 x := le_rfl,\n    have h1b : x \u2293 y \u2264 x := inf_le_left,\n    show x \u2294 (x \u2293 y) \u2264 x,\n      by exact sup_le h1a h1b,\n  },\n  have h2 : x \u2264 x \u2294 (x \u2293 y) := le_sup_left,\n  show x \u2294 (x \u2293 y) = x,\n    by exact le_antisymm h1 h2,\nend\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : x \u2294 (x \u2293 y) = x :=\nbegin\n  apply le_antisymm,\n  { apply sup_le,\n    { apply le_refl },\n    { apply inf_le_left }},\n  { apply le_sup_left },\nend\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : x \u2294 (x \u2293 y) = x :=\n-- by library_search\nsup_inf_self\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : x \u2294 (x \u2293 y) = x :=\n-- by hint\nby simp\n<\/pre>\n<p>Se puede interactuar con la prueba anterior en <a href=\"https:\/\/leanprover-community.github.io\/lean-web-editor\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus\/main\/src\/Ley_de_absorcion_2.lean\" rel=\"noopener noreferrer\" target=\"_blank\">esta sesi\u00f3n con Lean<\/a>.<\/p>\n<p><b>Referencias<\/b><\/p>\n<ul>\n<li>J. Avigad, K. Buzzard, R.Y. Lewis y P. Massot. <a href=\"https:\/\/bit.ly\/3U4UjBk\">Mathematics in Lean<\/a>, p. 22.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar que si R es un ret\u00edculo y x, y \u2208 R, entonces x \u2294 (x \u2293 y) = x Para ello, completar la siguiente teor\u00eda de Lean: import order.lattice variables {R : Type*} [lattice R] variables x y : R example : x \u2294 (x \u2293 y) = x := 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":[1],"tags":[293],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1141"}],"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=1141"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1141\/revisions"}],"predecessor-version":[{"id":1142,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1141\/revisions\/1142"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1141"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1141"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1141"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}