{"id":1145,"date":"2022-10-25T06:00:18","date_gmt":"2022-10-25T04:00:18","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1145"},"modified":"2022-10-21T11:17:57","modified_gmt":"2022-10-21T09:17:57","slug":"si-r-es-un-reticulo-tal-que-x-%e2%8a%94-y-%e2%8a%93-z-x-%e2%8a%94-y-%e2%8a%93-x-%e2%8a%94-z-entonces-a-%e2%8a%93-b-%e2%8a%94-c-a-%e2%8a%94-c-%e2%8a%93-b-%e2%8a%94-c","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/si-r-es-un-reticulo-tal-que-x-%e2%8a%94-y-%e2%8a%93-z-x-%e2%8a%94-y-%e2%8a%93-x-%e2%8a%94-z-entonces-a-%e2%8a%93-b-%e2%8a%94-c-a-%e2%8a%94-c-%e2%8a%93-b-%e2%8a%94-c\/","title":{"rendered":"Si R es un ret\u00edculo tal que x \u2294 (y \u2293 z) = (x \u2294 y) \u2293 (x \u2294 z), entonces (a \u2293 b) \u2294 c = (a \u2294 c) \u2293 (b \u2294 c)"},"content":{"rendered":"<p>Demostrar que Si R es un ret\u00edculo tal que<\/p>\n<pre lang=\"text\">\n   \u2200 x y z : \u03b1, x \u2294 (y \u2293 z) = (x \u2294 y) \u2293 (x \u2294 z)\n<\/pre>\n<p>entonces<\/p>\n<pre lang=\"text\">\n   \u2200 a b c : R, (a \u2293 b) \u2294 c = (a \u2294 c) \u2293 (b \u2294 c)\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]\n\nexample\n  (h : \u2200 x y z : R, x \u2294 (y \u2293 z) = (x \u2294 y) \u2293 (x \u2294 z))\n  : \u2200 a b c : R, (a \u2293 b) \u2294 c = (a \u2294 c) \u2293 (b \u2294 c) :=\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]\n\nexample\n  (h : \u2200 x y z : R, x \u2294 (y \u2293 z) = (x \u2294 y) \u2293 (x \u2294 z))\n  : \u2200 a b c : R, (a \u2293 b) \u2294 c = (a \u2294 c) \u2293 (b \u2294 c) :=\nbegin\n  intros a b c,\n  calc (a \u2293 b) \u2294 c\n       = c \u2294 (a \u2293 b)       : by rw sup_comm\n   ... = (c \u2294 a) \u2293 (c \u2294 b) : by rw h\n   ... = (a \u2294 c) \u2293 (c \u2294 b) : by rw [@sup_comm _ _ c a]\n   ... = (a \u2294 c) \u2293 (b \u2294 c) : by rw [@sup_comm _ _ c b]\nend\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\/Propiedad_distributiva_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. 23.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar que Si R es un ret\u00edculo tal que \u2200 x y z : \u03b1, x \u2294 (y \u2293 z) = (x \u2294 y) \u2293 (x \u2294 z) entonces \u2200 a b c : R, (a \u2293 b) \u2294 c = (a \u2294 c) \u2293 (b \u2294 c) Para ello, completar la siguiente teor\u00eda de Lean: import order.lattice variables {R : Type*} [lattice R] example (h : \u2200 x y z : R, x \u2294 (y \u2293 z) = (x \u2294 y) \u2293 (x \u2294 z)) : \u2200 a b c : R, (a \u2293 b) \u2294 c = (a \u2294 c) \u2293 (b \u2294 c) := 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\/1145"}],"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=1145"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1145\/revisions"}],"predecessor-version":[{"id":1146,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1145\/revisions\/1146"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1145"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1145"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1145"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}