{"id":1161,"date":"2022-10-31T06:00:21","date_gmt":"2022-10-31T04:00:21","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1161"},"modified":"2022-10-26T10:43:28","modified_gmt":"2022-10-26T08:43:28","slug":"si-x-es-un-espacio-metrico-y-x-y-%e2%88%88-x-entonces-distxy-%e2%89%a5-0","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/si-x-es-un-espacio-metrico-y-x-y-%e2%88%88-x-entonces-distxy-%e2%89%a5-0\/","title":{"rendered":"Si X es un espacio m\u00e9trico y x, y \u2208 X, entonces dist(x,y) \u2265 0"},"content":{"rendered":"<p>Demostrar que si X es un espacio m\u00e9trico y x, y \u2208 X, entonces<\/p>\n<pre lang=\"text\">\n   0 \u2264 dist x y\n<\/pre>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport topology.metric_space.basic\n\nvariables {X : Type*} [metric_space X]\nvariables x y : X\n\nexample : 0 \u2264 dist x y :=\nsorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Soluciones con Lean<\/b><\/p>\n<pre lang=\"lean\">\nimport topology.metric_space.basic\n\nvariables {X : Type*} [metric_space X]\nvariables x y : X\n\n-- 1\u00aa demostraci\u00f3n\nexample : 0 \u2264 dist x y :=\n  have h1 : 2 * dist x y \u2265 0, by calc\n    2 * dist x y = dist x y + dist x y : two_mul (dist x y)\n             ... = dist x y + dist y x : by rw [dist_comm x y]\n             ... \u2265 dist x x            : dist_triangle x y x\n             ... = 0                   : dist_self x,\n  show 0 \u2264 dist x y,\n    by exact nonneg_of_mul_nonneg_left h1 zero_lt_two\n\n-- 2\u00aa demostraci\u00f3n\nexample : 0 \u2264 dist x y :=\n-- by library_search\ndist_nonneg\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\/Distancia_no_negativa.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. 24.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar que si X es un espacio m\u00e9trico y x, y \u2208 X, entonces 0 \u2264 dist x y Para ello, completar la siguiente teor\u00eda de Lean: import topology.metric_space.basic variables {X : Type*} [metric_space X] variables x y : X example : 0 \u2264 dist x y := 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":[295],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1161"}],"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=1161"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1161\/revisions"}],"predecessor-version":[{"id":1162,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1161\/revisions\/1162"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1161"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1161"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1161"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}