{"id":949,"date":"2022-06-22T13:49:51","date_gmt":"2022-06-22T11:49:51","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=949"},"modified":"2022-06-22T13:56:52","modified_gmt":"2022-06-22T11:56:52","slug":"si-g%c2%b7f-es-inyectiva-entonces-f-es-inyectiva","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/si-g%c2%b7f-es-inyectiva-entonces-f-es-inyectiva\/","title":{"rendered":"Si g\u00b7f es inyectiva, entonces f es inyectiva."},"content":{"rendered":"<p>Demostrar que si g\u00b7f es inyectiva, entonces f es inyectiva.<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport tactic\nopen function\n\nvariables {X Y Z : Type}\nvariable  {f : X \u2192 Y}\nvariable  {g : Y \u2192 Z}\n\nexample\n  (Hgf : injective (g \u2218 f))\n  : injective f :=\nsorry\n<\/pre>\n<p><!--more--><\/p>\n<p><strong>Soluciones con Lean<\/strong><\/p>\n<pre lang=\"lean\">\nimport tactic\nopen function\n\nvariables {X Y Z : Type}\nvariable  {f : X \u2192 Y}\nvariable  {g : Y \u2192 Z}\n\n-- 1\u00aa demostraci\u00f3n\nexample\n  (Hgf : injective (g \u2218 f))\n  : injective f :=\nbegin\n  intros x x' f_xx',\n  apply Hgf,\n  calc (g \u2218 f) x = g (f x)    : rfl\n             ... = g (f x')   : congr_arg g f_xx'\n             ... = (g \u2218 f) x' : rfl\nend\n\n-- 2\u00aa demostraci\u00f3n\nexample\n  (Hgf : injective (g \u2218 f))\n  : injective f :=\nbegin\n  intros x x' f_xx',\n  apply Hgf,\n  simp [f_xx'],\nend\n\n-- 3\u00aa demostraci\u00f3n\nexample\n  (Hgf : injective (g \u2218 f))\n  : injective f :=\nbegin\n  intros x x' f_xx',\n  apply Hgf,\n  finish,\nend\n\n-- 4\u00aa demostraci\u00f3n\nexample\n  (Hgf : injective (g \u2218 f))\n  : injective f :=\nbegin\n  assume x  : X, \n  assume x' : X, \n  assume f_xx' : f x = f x',\n  have gf_xx' : (g \u2218 f) x = (g \u2218 f) x', from \n    calc (g \u2218 f) x = g (f x)    : rfl\n               ... = g (f x')   : congr_arg g f_xx'\n               ... = (g \u2218 f) x' : rfl,\n  show x = x', \n    { exact Hgf gf_xx' },\nend\n<\/pre>\n<p>El c\u00f3digo de las demostraciones se encuentra en <a href=\"https:\/\/github.com\/jaalonso\/Calculemus\/blob\/main\/src\/Si_gf_es_inyectiva_entonces_f_es_inyectiva.lean\">GitHub<\/a> y puede ejecutarse con el <a href=\"https:\/\/leanprover-community.github.io\/lean-web-editor\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus\/main\/src\/Si_gf_es_inyectiva_entonces_f_es_inyectiva.lean\">Lean Web editor<\/a>.<\/p>\n<p><strong>Soluciones con Isabelle\/HOL<\/strong><\/p>\n<pre lang=\"isar\">\ntheory La_composicion_de_funciones_inyectivas_es_inyectiva\nimports Main\nbegin\n\n(* 1\u00aa demostraci\u00f3n *)\nlemma\n  assumes \"inj f\"\n          \"inj g\"\n  shows   \"inj (f \u2218 g)\"\nproof (rule injI)\n  fix x y\n  assume \"(f \u2218 g) x = (f \u2218 g) y\"\n  then have \"f (g x) = f (g y)\"\n    by (simp only: o_apply)\n  then have \"g x = g y\"\n    using \u2039inj f\u203a by (simp only: injD)\n  then show \"x = y\" \n    using \u2039inj g\u203a by (simp only: injD)\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\nlemma\n  assumes \"inj f\"\n          \"inj g\"\n  shows   \"inj (f \u2218 g)\"\nusing assms\nby (simp add: inj_def)\n\n(* 3\u00aa demostraci\u00f3n *)\nlemma\n  assumes \"inj f\"\n          \"inj g\"\n  shows   \"inj (f \u2218 g)\"\nusing assms\nby (rule inj_compose)\n\nend\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar que si g\u00b7f es inyectiva, entonces f es inyectiva. Para ello, completar la siguiente teor\u00eda de Lean: import tactic open function variables {X Y Z : Type} variable {f : X \u2192 Y} variable {g : Y \u2192 Z} example (Hgf : injective (g \u2218 f)) : injective f := 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":[7],"tags":[282],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/949"}],"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=949"}],"version-history":[{"count":4,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/949\/revisions"}],"predecessor-version":[{"id":953,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/949\/revisions\/953"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=949"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=949"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=949"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}