{"id":1767,"date":"2023-11-13T06:00:03","date_gmt":"2023-11-13T04:00:03","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1767"},"modified":"2023-11-02T13:31:24","modified_gmt":"2023-11-02T11:31:24","slug":"13-nov-23","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/13-nov-23\/","title":{"rendered":"Si f: \u211d \u2192 \u211d es suprayectiva, entonces \u2203x \u2208 \u211d tal que f(x)\u00b2 = 9"},"content":{"rendered":"<p>Demostrar con Lean4 que si &#92;(f: \u211d \u2192 \u211d&#92;) es suprayectiva, entonces &#92;(\u2203x \u2208 \u211d&#92;) tal que &#92;(f(x)\u00b2 = 9&#92;).<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Real.Basic\nopen Function\n\nexample\n  {f : \u211d \u2192 \u211d}\n  (h : Surjective f)\n  : \u2203 x, (f x)^2 = 9 :=\nby sorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Demostraci\u00f3n en lenguaje natural<\/b><\/p>\n<p><br \/>\nAl ser &#92;(f&#92;) suprayectiva, existe un &#92;(y&#92;) tal que &#92;(f(y) = 3&#92;). Por tanto, &#92;(f(y)\u00b2 = 9&#92;).<\/p>\n<p><b>Demostraciones con Lean4<\/b><\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Real.Basic\n\nopen Function\n\nexample\n  {f : \u211d \u2192 \u211d}\n  (h : Surjective f)\n  : \u2203 x, (f x)^2 = 9 :=\nby\n  cases' h 3 with y hy\n  -- y : \u211d\n  -- hy : f y = 3\n  use y\n  -- \u22a2 f y ^ 2 = 9\n  rw [hy]\n  -- \u22a2 3 ^ 2 = 9\n  norm_num\n<\/pre>\n<p><b>Demostraciones interactivas<\/b><\/p>\n<p>Se puede interactuar con las demostraciones anteriores en <a href=\"https:\/\/live.lean-lang.org\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus2\/main\/src\/Propiedad_de_suprayectivas.lean\" rel=\"noopener noreferrer\" target=\"_blank\">Lean 4 Web<\/a>.<\/p>\n<p><b>Referencias<\/b><\/p>\n<ul>\n<li> J. Avigad y P. Massot. <a href=\"https:\/\/bit.ly\/3U4UjBk\">Mathematics in Lean<\/a>, p. 31.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar con Lean4 que si &#92;(f: \u211d \u2192 \u211d&#92;) es suprayectiva, entonces &#92;(\u2203x \u2208 \u211d&#92;) tal que &#92;(f(x)\u00b2 = 9&#92;). Para ello, completar la siguiente teor\u00eda de Lean4: import Mathlib.Data.Real.Basic open Function example {f : \u211d \u2192 \u211d} (h : Surjective f) : \u2203 x, (f x)^2 = 9 := by 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\/1767"}],"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=1767"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1767\/revisions"}],"predecessor-version":[{"id":1768,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1767\/revisions\/1768"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1767"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1767"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1767"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}