{"id":407,"date":"2021-06-15T06:00:44","date_gmt":"2021-06-15T04:00:44","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=407"},"modified":"2021-06-12T13:52:18","modified_gmt":"2021-06-12T11:52:18","slug":"imagen-de-la-interseccion","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/imagen-de-la-interseccion\/","title":{"rendered":"Imagen de la intersecci\u00f3n"},"content":{"rendered":"

Demostrar que<\/p>\n

\n   f[s \u2229 t] \u2286 f[s] \u2229 f[t]\n<\/pre>\n
Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n
\nimport data.set.basic\nimport tactic\n\nopen set\n\nvariables {\u03b1 : Type*} {\u03b2 : Type*}\nvariable  f : \u03b1 \u2192 \u03b2\nvariables s t : set \u03b1\n\nexample : f '' (s \u2229 t) \u2286 f '' s \u2229 f '' t :=\nsorry\n<\/pre>\n
[expand title=\u00bbSoluciones con Lean\u00bb]<\/p>\n
\r\nimport data.set.basic\r\nimport tactic\r\n\r\nopen set\r\n\r\nvariables {\u03b1 : Type*} {\u03b2 : Type*}\r\nvariable  f : \u03b1 \u2192 \u03b2\r\nvariables s t : set \u03b1\r\n\r\n-- 1\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample : f '' (s \u2229 t) \u2286 f '' s \u2229 f '' t :=\r\nbegin\r\n  intros y hy,\r\n  cases hy with x hx,\r\n  cases hx with xst fxy,\r\n  split,\r\n  { use x,\r\n    split,\r\n    { exact xst.1, },\r\n    { exact fxy, }},\r\n  { use x,\r\n    split,\r\n    { exact xst.2, },\r\n    { exact fxy, }},\r\nend\r\n\r\n-- 2\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample : f '' (s \u2229 t) \u2286 f '' s \u2229 f '' t :=\r\nbegin\r\n  intros y hy,\r\n  rcases hy with \u27e8x, \u27e8xs, xt\u27e9, fxy\u27e9,\r\n  split,\r\n  { use x,\r\n    exact \u27e8xs, fxy\u27e9, },\r\n  { use x,\r\n    exact \u27e8xt, fxy\u27e9, },\r\nend\r\n\r\n-- 3\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample : f '' (s \u2229 t) \u2286 f '' s \u2229 f '' t :=\r\nbegin\r\n  rintros y \u27e8x, \u27e8xs, xt\u27e9, fxy\u27e9,\r\n  split,\r\n  { use [x, xs, fxy], },\r\n  { use [x, xt, fxy], },\r\nend\r\n\r\n-- 4\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample : f '' (s \u2229 t) \u2286 f '' s \u2229 f '' t :=\r\nimage_inter_subset f s t\r\n\r\n-- 5\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample : f '' (s \u2229 t) \u2286 f '' s \u2229 f '' t :=\r\nby intro ; finish\r\n<\/pre>\n
Se puede interactuar con la prueba anterior en esta sesi\u00f3n con Lean<\/a>,
\n[\/expand]<\/p>\n
[expand title=\u00bbSoluciones con Isabelle\/HOL\u00bb]<\/p>\n
\r\ntheory Imagen_de_la_interseccion\r\nimports Main\r\nbegin\r\n\r\nsection \u20391\u00aa demostraci\u00f3n\u203a\r\n\r\nlemma \"f ` (s \u2229 t) \u2286 f ` s \u2229 f ` t\"\r\nproof (rule subsetI)\r\n  fix y\r\n  assume \"y \u2208 f ` (s \u2229 t)\"\r\n  then have \"y \u2208 f ` s\"\r\n  proof (rule imageE)\r\n    fix x\r\n    assume \"y = f x\"\r\n    assume \"x \u2208 s \u2229 t\"\r\n    have \"x \u2208 s\"\r\n      using \u2039x \u2208 s \u2229 t\u203a by (rule IntD1)\r\n    then have \"f x \u2208 f ` s\"\r\n      by (rule imageI)\r\n    with \u2039y = f x\u203a show \"y \u2208 f ` s\"\r\n      by (rule ssubst)\r\n  qed\r\n  moreover\r\n  note \u2039y \u2208 f ` (s \u2229 t)\u203a\r\n  then have \"y \u2208 f ` t\"\r\n  proof (rule imageE)\r\n    fix x\r\n    assume \"y = f x\"\r\n    assume \"x \u2208 s \u2229 t\"\r\n    have \"x \u2208 t\"\r\n      using \u2039x \u2208 s \u2229 t\u203a by (rule IntD2)\r\n    then have \"f x \u2208 f ` t\"\r\n      by (rule imageI)\r\n    with \u2039y = f x\u203a show \"y \u2208 f ` t\"\r\n      by (rule ssubst)\r\n  qed\r\n  ultimately show \"y \u2208 f ` s \u2229 f ` t\"\r\n    by (rule IntI)\r\nqed\r\n\r\nsection \u20392\u00aa demostraci\u00f3n\u203a\r\n\r\nlemma \"f ` (s \u2229 t) \u2286 f ` s \u2229 f ` t\"\r\nproof\r\n  fix y\r\n  assume \"y \u2208 f ` (s \u2229 t)\"\r\n  then have \"y \u2208 f ` s\"\r\n  proof\r\n    fix x\r\n    assume \"y = f x\"\r\n    assume \"x \u2208 s \u2229 t\"\r\n    have \"x \u2208 s\"\r\n      using \u2039x \u2208 s \u2229 t\u203a by simp\r\n    then have \"f x \u2208 f ` s\"\r\n      by simp\r\n    with \u2039y = f x\u203a show \"y \u2208 f ` s\"\r\n      by simp\r\n  qed\r\n  moreover\r\n  note \u2039y \u2208 f ` (s \u2229 t)\u203a\r\n  then have \"y \u2208 f ` t\"\r\n  proof\r\n    fix x\r\n    assume \"y = f x\"\r\n    assume \"x \u2208 s \u2229 t\"\r\n    have \"x \u2208 t\"\r\n      using \u2039x \u2208 s \u2229 t\u203a by simp\r\n    then have \"f x \u2208 f ` t\"\r\n      by simp\r\n    with \u2039y = f x\u203a show \"y \u2208 f ` t\"\r\n      by simp\r\n  qed\r\n  ultimately show \"y \u2208 f ` s \u2229 f ` t\"\r\n    by simp\r\nqed\r\n\r\nsection \u20393\u00aa demostraci\u00f3n\u203a\r\n\r\nlemma \"f ` (s \u2229 t) \u2286 f ` s \u2229 f ` t\"\r\nproof\r\n  fix y\r\n  assume \"y \u2208 f ` (s \u2229 t)\"\r\n  then obtain x where hx : \"y = f x \u2227 x \u2208 s \u2229 t\" by auto\r\n  then have \"y = f x\" by simp\r\n  have \"x \u2208 s\" using hx by simp\r\n  have \"x \u2208 t\" using hx by simp\r\n  have \"y \u2208  f ` s\" using \u2039y = f x\u203a \u2039x \u2208 s\u203a by simp\r\n  moreover\r\n  have \"y \u2208  f ` t\" using \u2039y = f x\u203a \u2039x \u2208 t\u203a by simp\r\n  ultimately show \"y \u2208 f ` s \u2229 f ` t\"\r\n    by simp\r\nqed\r\n\r\nsection \u20394\u00aa demostraci\u00f3n\u203a\r\n\r\nlemma \"f ` (s \u2229 t) \u2286 f ` s \u2229 f ` t\"\r\n  by (simp only: image_Int_subset)\r\n\r\nsection \u20395\u00aa demostraci\u00f3n\u203a\r\n\r\nlemma \"f ` (s \u2229 t) \u2286 f ` s \u2229 f ` t\"\r\n  by auto\r\n\r\nend\r\n<\/pre>\n
[expand title=\u00bbNuevas soluciones\u00bb]<\/p>\n
    \n
  • En los comentarios se pueden escribir nuevas soluciones.\n
  • El c\u00f3digo se debe escribir entre una l\u00ednea con <pre lang="lean"> (o <pre lang="isar">) y otra con <\/pre>\n<\/ul>\n
    [\/expand]<\/p>\n","protected":false},"excerpt":{"rendered":"
    Demostrar que f[s \u2229 t] \u2286 f[s] \u2229 f[t] Para ello, completar la siguiente teor\u00eda de Lean: import data.set.basic import tactic open set variables {\u03b1 : Type*} {\u03b2 : Type*} variable f : \u03b1 \u2192 \u03b2 variables s t : set \u03b1 example : f \u00bb (s \u2229 t) \u2286 f \u00bb s \u2229 f \u00bb t := sorry [expand title=\u00bbSoluciones con Lean\u00bb] import data.set.basic import tactic open set variables {\u03b1 : Type*} {\u03b2 : Type*} variable f : \u03b1 \u2192 \u03b2 variables s t : set \u03b1 — 1\u00aa demostraci\u00f3n — =============== example : f \u00bb (s \u2229 t) \u2286 f \u00bb s \u2229 f \u00bb t := begin intros y hy, cases hy with x hx, cases hx with xst fxy, split, {…<\/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":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/407"}],"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=407"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/407\/revisions"}],"predecessor-version":[{"id":476,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/407\/revisions\/476"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=407"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=407"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=407"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}