{"id":1212,"date":"2022-11-22T09:32:24","date_gmt":"2022-11-22T07:32:24","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1212"},"modified":"2022-11-22T09:34:15","modified_gmt":"2022-11-22T07:34:15","slug":"22-nov-22","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/22-nov-22\/","title":{"rendered":"Para todo c \u2208 \u211d, la funci\u00f3n f(x) = x+c es inyectiva"},"content":{"rendered":"<p>Demostrar que para todo c \u2208 \u211d, la funci\u00f3n f(x) = x+c es inyectiva.<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport data.real.basic                                               \n\nopen function                                                        \n\nvariable {c : \u211d}\n\nexample : injective (\u03bb x, x + c) :=\nsorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Soluciones con Lean<\/b><\/p>\n<pre lang=\"lean\">\nimport data.real.basic                                               \n\nopen function                                                        \nvariable {c : \u211d}\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : injective (\u03bb x, x + c) :=\nbegin\n  assume x1 : \u211d,\n  assume x2 : \u211d,\n  assume h1 : (\u03bb x, x + c) x1 = (\u03bb x, x + c) x2,\n  have h2 : x1 + c = x2 + c := h1,\n  show x1 = x2,\n    by exact (add_left_inj c).mp h2,\nend\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : injective (\u03bb x, x + c) :=\nbegin\n  intros x1 x2 h,\n  change x1 + c = x2 + c at h,\n  apply add_right_cancel h,\nend\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : injective (\u03bb x, x + c) :=\nbegin\n  intros x1 x2 h,\n  apply (add_left_inj c).mp,\n  exact h,\nend\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : injective (\u03bb x, x + c) :=\n\u03bb x1 x2 h, (add_left_inj c).mp h\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\/Suma_constante_es_inyectiva.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 para todo c \u2208 \u211d, la funci\u00f3n f(x) = x+c es inyectiva. Para ello, completar la siguiente teor\u00eda de Lean: import data.real.basic open function variable {c : \u211d} example : injective (\u03bb x, x + c) := 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\/1212"}],"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=1212"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1212\/revisions"}],"predecessor-version":[{"id":1215,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1212\/revisions\/1215"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1212"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1212"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1212"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}