{"id":633,"date":"2021-08-06T06:00:37","date_gmt":"2021-08-06T04:00:37","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=633"},"modified":"2021-08-03T17:03:48","modified_gmt":"2021-08-03T15:03:48","slug":"las-funciones-suprayectivas-tienen-inversa-por-la-derecha","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/las-funciones-suprayectivas-tienen-inversa-por-la-derecha\/","title":{"rendered":"Las funciones suprayectivas tienen inversa por la derecha"},"content":{"rendered":"

En Lean, que g es una inversa por la izquierda de f est\u00e1 definido por<\/p>\n

\n   left_inverse (g : \u03b2 \u2192 \u03b1) (f : \u03b1 \u2192 \u03b2) : Prop :=\n      \u2200 x, g (f x) = x\n<\/pre>\n
que g es una inversa por la derecha de f est\u00e1 definido por<\/p>\n
\n   right_inverse (g : \u03b2 \u2192 \u03b1) (f : \u03b1 \u2192 \u03b2) : Prop :=\n      left_inverse f g\n<\/pre>\n
y que f tenga inversa por la derecha est\u00e1 definido por<\/p>\n
\n   has_right_inverse (f : \u03b1 \u2192 \u03b2) : Prop :=\n      \u2203 g : \u03b2 \u2192 \u03b1, right_inverse g f\n<\/pre>\n
Finalmente, que f es suprayectiva est\u00e1 definido por<\/p>\n
\n   def surjective (f : \u03b1 \u2192 \u03b2) : Prop :=\n      \u2200 b, \u2203 a, f a = b\n<\/pre>\n
Demostrar que si f es una funci\u00f3n suprayectiva, entonces f tiene inversa por la derecha.<\/p>\n
Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n
\nimport tactic\nopen function classical\n\nvariables {\u03b1 \u03b2: Type*}\nvariable  {f : \u03b1 \u2192 \u03b2}\n\nexample\n  (hf : surjective f)\n  : has_right_inverse f :=\nsorry\n<\/pre>\n
[expand title=\u00bbSoluciones con Lean\u00bb]<\/p>\n
\r\nimport tactic\r\nopen function classical\r\n\r\nvariables {\u03b1 \u03b2: Type*}\r\nvariable  {f : \u03b1 \u2192 \u03b2}\r\n\r\n-- 1\u00aa demostraci\u00f3n\r\nexample\r\n  (hf : surjective f)\r\n  : has_right_inverse f :=\r\nbegin\r\n  unfold has_right_inverse,\r\n  let g := \u03bb y, some (hf y),\r\n  use g,\r\n  unfold function.right_inverse,\r\n  unfold function.left_inverse,\r\n  intro b,\r\n  apply some_spec (hf b),\r\nend\r\n\r\n-- 2\u00aa demostraci\u00f3n\r\nexample\r\n  (hf : surjective f)\r\n  : has_right_inverse f :=\r\nbegin\r\n  let g := \u03bb y, some (hf y),\r\n  use g,\r\n  intro b,\r\n  apply some_spec (hf b),\r\nend\r\n\r\n-- 3\u00aa demostraci\u00f3n\r\nexample\r\n  (hf : surjective f)\r\n  : has_right_inverse f :=\r\nbegin\r\n  use surj_inv hf,\r\n  intro b,\r\n  exact surj_inv_eq hf b,\r\nend\r\n\r\n-- 4\u00aa demostraci\u00f3n\r\nexample\r\n  (hf : surjective f)\r\n  : has_right_inverse f :=\r\nbegin\r\n  use surj_inv hf,\r\n  exact surj_inv_eq hf,\r\nend\r\n\r\n-- 5\u00aa demostraci\u00f3n\r\nexample\r\n  (hf : surjective f)\r\n  : has_right_inverse f :=\r\nbegin\r\n  use [surj_inv hf, surj_inv_eq hf],\r\nend\r\n\r\n-- 6\u00aa demostraci\u00f3n\r\nexample\r\n  (hf : surjective f)\r\n  : has_right_inverse f :=\r\n\u27e8surj_inv hf, surj_inv_eq hf\u27e9\r\n\r\n-- 7\u00aa demostraci\u00f3n\r\nexample\r\n  (hf : surjective f)\r\n  : has_right_inverse f :=\r\n\u27e8_, surj_inv_eq hf\u27e9\r\n\r\n-- 8\u00aa demostraci\u00f3n\r\nexample\r\n  (hf : surjective f)\r\n  : has_right_inverse f :=\r\nsurjective.has_right_inverse hf\r\n<\/pre>\n
Se puede interactuar con la prueba anterior en esta sesi\u00f3n con Lean<\/a>.<\/p>\n
En los comentarios se pueden escribir otras soluciones, escribiendo el c\u00f3digo entre una l\u00ednea con <pre lang="lean"> y otra con <\/pre>
\n[\/expand]<\/p>\n
[expand title=\u00bbSoluciones con Isabelle\/HOL\u00bb]<\/p>\n
\r\ntheory Las_funciones_suprayectivas_tienen_inversa_por_la_derecha\r\nimports Main\r\nbegin\r\n\r\ndefinition tiene_inversa_dcha :: \"('a \u21d2 'b) \u21d2 bool\" where\r\n  \"tiene_inversa_dcha f \u27f7 (\u2203g. \u2200y. f (g y) = y)\"\r\n\r\n(* 1\u00aa demostraci\u00f3n *)\r\nlemma\r\n  assumes \"surj f\"\r\n  shows   \"tiene_inversa_dcha f\"\r\nproof (unfold tiene_inversa_dcha_def)\r\n  let ?g = \"\u03bby. SOME x. f x = y\"\r\n  have \"\u2200y. f (?g y) = y\"\r\n  proof (rule allI)\r\n    fix y\r\n    have \"\u2203x. y = f x\"\r\n      using assms by (rule surjD)\r\n    then have \"\u2203x. f x = y\"\r\n      by auto\r\n    then show \"f (?g y) = y\"\r\n      by (rule someI_ex)\r\n  qed\r\n  then show \"\u2203g. \u2200y. f (g y) = y\"\r\n    by auto\r\nqed\r\n\r\n(* 2\u00aa demostraci\u00f3n *)\r\nlemma\r\n  assumes \"surj f\"\r\n  shows   \"tiene_inversa_dcha f\"\r\nproof (unfold tiene_inversa_dcha_def)\r\n  let ?g = \"\u03bby. SOME x. f x = y\"\r\n  have \"\u2200y. f (?g y) = y\"\r\n  proof (rule allI)\r\n    fix y\r\n    have \"\u2203x. f x = y\"\r\n      by (metis assms surjD)\r\n    then show \"f (?g y) = y\"\r\n      by (rule someI_ex)\r\n  qed\r\n  then show \"\u2203g. \u2200y. f (g y) = y\"\r\n    by auto\r\nqed\r\n\r\n(* 3\u00aa demostraci\u00f3n *)\r\nlemma\r\n  assumes \"surj f\"\r\n  shows   \"tiene_inversa_dcha f\"\r\nproof (unfold tiene_inversa_dcha_def)\r\n  have \"\u2200y. f (inv f y) = y\"\r\n    by (simp add: assms surj_f_inv_f)\r\n  then show \"\u2203g. \u2200y. f (g y) = y\"\r\n    by auto\r\nqed\r\n\r\n(* 4\u00aa demostraci\u00f3n *)\r\nlemma\r\n  assumes \"surj f\"\r\n  shows   \"tiene_inversa_dcha f\"\r\n  by (metis assms surjD tiene_inversa_dcha_def)\r\n\r\nend\r\n<\/pre>\n
En los comentarios se pueden escribir otras soluciones, escribiendo el c\u00f3digo entre una l\u00ednea con <pre lang="isar"> y otra con <\/pre>
\n[\/expand]<\/p>\n","protected":false},"excerpt":{"rendered":"
En Lean, que g es una inversa por la izquierda de f est\u00e1 definido por left_inverse (g : \u03b2 \u2192 \u03b1) (f : \u03b1 \u2192 \u03b2) : Prop := \u2200 x, g (f x) = x que g es una inversa por la derecha de f est\u00e1 definido por right_inverse (g : \u03b2 \u2192 \u03b1) (f : \u03b1 \u2192 \u03b2) : Prop := left_inverse f g y que f tenga inversa por la derecha est\u00e1 definido por has_right_inverse (f : \u03b1 \u2192 \u03b2) : Prop := \u2203 g : \u03b2 \u2192 \u03b1, right_inverse g f Finalmente, que f es suprayectiva est\u00e1 definido por def surjective (f : \u03b1 \u2192 \u03b2) : Prop := \u2200 b, \u2203 a, f a = b Demostrar que si f…<\/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":[17],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/633"}],"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=633"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/633\/revisions"}],"predecessor-version":[{"id":634,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/633\/revisions\/634"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=633"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=633"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=633"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}