{"id":1040,"date":"2022-09-20T06:00:20","date_gmt":"2022-09-20T04:00:20","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1040"},"modified":"2022-09-11T16:36:21","modified_gmt":"2022-09-11T14:36:21","slug":"si-a-b-c-d-e-%e2%88%88-%e2%84%9d-tales-que-a-%e2%89%a4-b-b-c-c-%e2%89%a4-d-d-e-entonces-a-e","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/si-a-b-c-d-e-%e2%88%88-%e2%84%9d-tales-que-a-%e2%89%a4-b-b-c-c-%e2%89%a4-d-d-e-entonces-a-e\/","title":{"rendered":"Si a, b, c, d, e \u2208 \u211d tales que a \u2264 b, b < c, c \u2264 d, d < e, entonces a < e"},"content":{"rendered":"<p>Demostrar que si a, b, c, d, e \u2208 \u211d tales que<\/p>\n<pre lang=\"text\">\na \u2264 b\nb < c\nc \u2264 d\nd < e\n<\/pre>\n<p>entonces<\/p>\n<pre lang=\"text\">\na < e\n<\/pre>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport data.real.basic\n\nvariables a b c d e : \u211d\n\nexample\n  (h\u2080 : a \u2264 b)\n  (h\u2081 : b < c)\n  (h\u2082 : c \u2264 d)\n  (h\u2083 : d < e) :\n  a < e :=\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 c d e : \u211d\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h\u2080 : a \u2264 b)\n  (h\u2081 : b < c)\n  (h\u2082 : c \u2264 d)\n  (h\u2083 : d < e) :\n  a < e :=\ncalc a \u2264 b : h\u2080\n   ... < c : h\u2081\n   ... \u2264 d : h\u2082\n   ... < e : h\u2083\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h\u2080 : a \u2264 b)\n  (h\u2081 : b < c)\n  (h\u2082 : c \u2264 d)\n  (h\u2083 : d < e) :\n  a < e :=\nbegin\n  apply lt_of_le_of_lt h\u2080,\n  apply lt_trans h\u2081,\n  apply lt_of_le_of_lt h\u2082,\n  exact h\u2083,\nend\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h\u2080 : a \u2264 b)\n  (h\u2081 : b < c)\n  (h\u2082 : c \u2264 d)\n  (h\u2083 : d < e) :\n  a < e :=\nby finish\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h\u2080 : a \u2264 b)\n  (h\u2081 : b < c)\n  (h\u2082 : c \u2264 d)\n  (h\u2083 : d < e) :\n  a < e :=\nby linarith\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\/Ejercicio_sobre_orden.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>.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar que si a, b, c, d, e \u2208 \u211d tales que a \u2264 b b < c c \u2264 d d < e entonces a < e Para ello, completar la siguiente teor\u00eda de Lean: import data.real.basic variables a b c d e : \u211d example (h\u2080 : a \u2264 b) (h\u2081 : b < c) (h\u2082 : c \u2264 d) (h\u2083 : d < e) : a < e := sorry\n<\/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":[286,287],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1040"}],"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=1040"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1040\/revisions"}],"predecessor-version":[{"id":1101,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1040\/revisions\/1101"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1040"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1040"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1040"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}