{"id":1206,"date":"2022-11-19T06:00:54","date_gmt":"2022-11-19T04:00:54","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1206"},"modified":"2022-11-18T17:50:13","modified_gmt":"2022-11-18T15:50:13","slug":"19-nov-22","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/19-nov-22\/","title":{"rendered":"Si r \u2286 s y s \u2286 t, entonces r \u2286 t"},"content":{"rendered":"<p>Demostrar que si r \u2286 s y s \u2286 t, entonces r \u2286 t.<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport tactic\n\nvariables {\u03b1 : Type*}\nvariables r s t : set \u03b1\n\nexample\n  (rs : r \u2286 s)\n  (st : s \u2286 t)\n  : r \u2286 t :=\nsorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Soluciones con Lean<\/b><\/p>\n<pre lang=\"lean\">\nimport tactic\n\nvariables {\u03b1 : Type*}\nvariables r s t : set \u03b1\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (rs : r \u2286 s)\n  (st : s \u2286 t)\n  : r \u2286 t :=\nbegin\n  assume x,\n  assume xr : x \u2208 r,\n  have h1 : x \u2208 s := rs xr,\n  show x \u2208 t,\n    by exact st h1,\nend\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (rs : r \u2286 s)\n  (st : s \u2286 t)\n  : r \u2286 t :=\nbegin\n  intros x xr,\n  apply st,\n  apply rs,\n  exact xr\nend\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (rs : r \u2286 s)\n  (st : s \u2286 t)\n  : r \u2286 t :=\n\u03bb x xr, st (rs xr)\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (rs : r \u2286 s)\n  (st : s \u2286 t)\n  : r \u2286 t :=\n-- by library_search\nset.subset.trans rs st\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (rs : r \u2286 s)\n  (st : s \u2286 t)\n  : r \u2286 t :=\n-- by hint\nby tauto\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_transitiva_del_subconjunto.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. 29.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar que si r \u2286 s y s \u2286 t, entonces r \u2286 t. Para ello, completar la siguiente teor\u00eda de Lean: import tactic variables {\u03b1 : Type*} variables r s t : set \u03b1 example (rs : r \u2286 s) (st : s \u2286 t) : r \u2286 t := 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":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1206"}],"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=1206"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1206\/revisions"}],"predecessor-version":[{"id":1209,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1206\/revisions\/1209"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1206"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1206"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1206"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}