{"id":1063,"date":"2022-09-29T06:00:29","date_gmt":"2022-09-29T04:00:29","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1063"},"modified":"2022-09-11T16:44:43","modified_gmt":"2022-09-11T14:44:43","slug":"si-a-b-%e2%88%88-%e2%84%9d-entonces-minab-minba","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/si-a-b-%e2%88%88-%e2%84%9d-entonces-minab-minba\/","title":{"rendered":"Si a, b \u2208 \u211d, entonces min(a,b) = min(b,a)"},"content":{"rendered":"<p>Demostrar que si a, b \u2208 \u211d, entonces min(a,b) = min(b,a).<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport data.real.basic\n\nvariables a b : \u211d\n\nexample : min a b = min b a :=\nsorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Soluciones con Lean<\/b><\/p>\n<pre lang=\"lean\">\nimport data.real.basic\n\nvariables a b : \u211d\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : min a b = min b a :=\nbegin\n  apply le_antisymm,\n  { show min a b \u2264 min b a,\n    apply le_min,\n    { apply min_le_right },\n    { apply min_le_left }},\n  { show min b a \u2264 min a b,\n    apply le_min,\n    { apply min_le_right },\n    { apply min_le_left }},\nend\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : min a b = min b a :=\nbegin\n  have h : \u2200 x y : \u211d, min x y \u2264 min y x,\n  { intros x y,\n    apply le_min,\n    { apply min_le_right },\n    { apply min_le_left }},\n  apply le_antisymm,\n  apply h,\n  apply h,\nend\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : min a b = min b a :=\nbegin\n  have h : \u2200 {x y : \u211d}, min x y \u2264 min y x,\n  { intros x y,\n    exact le_min (min_le_right x y) (min_le_left x y) },\n  exact le_antisymm h h,\nend\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : min a b = min b a :=\nbegin\n  apply le_antisymm,\n  repeat {\n    apply le_min,\n    apply min_le_right,\n    apply min_le_left },\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\/Conmutatividad_del_minimo.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. 19.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar que si a, b \u2208 \u211d, entonces min(a,b) = min(b,a). Para ello, completar la siguiente teor\u00eda de Lean: import data.real.basic variables a b : \u211d example : min a b = min b a := 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":[286],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1063"}],"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=1063"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1063\/revisions"}],"predecessor-version":[{"id":1108,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1063\/revisions\/1108"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1063"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1063"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1063"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}