{"id":485,"date":"2021-06-20T06:00:40","date_gmt":"2021-06-20T04:00:40","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=485"},"modified":"2021-06-14T12:57:20","modified_gmt":"2021-06-14T10:57:20","slug":"union-con-la-imagen","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/union-con-la-imagen\/","title":{"rendered":"Uni\u00f3n con la imagen"},"content":{"rendered":"<p>Demostrar que<\/p>\n<pre lang=\"text\">\n   f[s \u222a f\u207b\u00b9[v]] \u2286 f[s] \u222a v\n<\/pre>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport data.set.basic\nimport tactic\n\nopen set\n\nvariables {\u03b1 : Type*} {\u03b2 : Type*}\nvariable  f : \u03b1 \u2192 \u03b2\nvariable  s : set \u03b1\nvariable  v : set \u03b2\n\nexample : f '' (s \u222a f \u207b\u00b9' v) \u2286 f '' s \u222a v :=\nsorry\n<\/pre>\n<p>[expand title=\u00bbSoluciones con Lean\u00bb]<\/p>\n<pre lang=\"lean\">\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\nvariable  s : set \u03b1\r\nvariable  v : set \u03b2\r\n\r\n-- 1\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample : f '' (s \u222a f \u207b\u00b9' v) \u2286 f '' s \u222a v :=\r\nbegin\r\n  intros y hy,\r\n  cases hy with x hx,\r\n  cases hx with hx1 fxy,\r\n  cases hx1 with xs xv,\r\n  { left,\r\n    use x,\r\n    split,\r\n    { exact xs, },\r\n    { exact fxy, }},\r\n  { right,\r\n    rw \u2190 fxy,\r\n    exact xv, },\r\nend\r\n\r\n-- 2\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample : f '' (s \u222a f \u207b\u00b9' v) \u2286 f '' s \u222a v :=\r\nbegin\r\n  rintros y \u27e8x, xs | xv, fxy\u27e9,\r\n  { left,\r\n    use [x, xs, fxy], },\r\n  { right,\r\n    rw \u2190 fxy,\r\n    exact xv, },\r\nend\r\n\r\n-- 3\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample : f '' (s \u222a f \u207b\u00b9' v) \u2286 f '' s \u222a v :=\r\nbegin\r\n  rintros y \u27e8x, xs | xv, fxy\u27e9;\r\n  finish,\r\nend\r\n<\/pre>\n<p>Se puede interactuar con la prueba anterior en <a href=\"https:\/\/www.cs.us.es\/~jalonso\/lean-web-editor\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus\/main\/src\/Union_con_la_imagen.lean\" rel=\"noopener noreferrer\" target=\"_blank\">esta sesi\u00f3n con Lean<\/a>,<\/p>\n<p>En los comentarios se pueden escribir otras soluciones, escribiendo el c\u00f3digo entre una l\u00ednea con &#60;pre lang=&quot;isar&quot;&#62; y otra con &#60;\/pre&#62;<br \/>\n[\/expand]<\/p>\n<p>[expand title=\u00bbSoluciones con Isabelle\/HOL\u00bb]<\/p>\n<pre lang=\"isar\">\r\ntheory Union_con_la_imagen\r\nimports Main\r\nbegin\r\n\r\n(* 1\u00aa demostraci\u00f3n *)\r\n\r\nlemma \"f ` (s \u222a f -` v) \u2286 f ` s \u222a v\"\r\nproof (rule subsetI)\r\n  fix y\r\n  assume \"y \u2208 f ` (s \u222a f -` v)\"\r\n  then show \"y \u2208 f ` s \u222a v\"\r\n  proof (rule imageE)\r\n    fix x\r\n    assume \"y = f x\"\r\n    assume \"x \u2208 s \u222a f -` v\"\r\n    then show \"y \u2208 f ` s \u222a v\"\r\n    proof (rule UnE)\r\n      assume \"x \u2208 s\"\r\n      then have \"f x \u2208 f ` s\"\r\n        by (rule imageI)\r\n      with \u2039y = f x\u203a have \"y \u2208 f ` s\"\r\n        by (rule ssubst)\r\n      then show \"y \u2208 f ` s \u222a v\"\r\n        by (rule UnI1)\r\n    next\r\n      assume \"x \u2208 f -` v\"\r\n      then have \"f x \u2208 v\"\r\n        by (rule vimageD)\r\n      with \u2039y = f x\u203a have \"y \u2208 v\"\r\n        by (rule ssubst)\r\n      then show \"y \u2208 f ` s \u222a v\"\r\n        by (rule UnI2)\r\n    qed\r\n  qed\r\nqed\r\n\r\n(* 2\u00aa demostraci\u00f3n *)\r\n\r\nlemma \"f ` (s \u222a f -` v) \u2286 f ` s \u222a v\"\r\nproof\r\n  fix y\r\n  assume \"y \u2208 f ` (s \u222a f -` v)\"\r\n  then show \"y \u2208 f ` s \u222a v\"\r\n  proof\r\n    fix x\r\n    assume \"y = f x\"\r\n    assume \"x \u2208 s \u222a f -` v\"\r\n    then show \"y \u2208 f ` s \u222a v\"\r\n    proof\r\n      assume \"x \u2208 s\"\r\n      then have \"f x \u2208 f ` s\" by simp\r\n      with \u2039y = f x\u203a have \"y \u2208 f ` s\" by simp\r\n      then show \"y \u2208 f ` s \u222a v\" by simp\r\n    next\r\n      assume \"x \u2208 f -` v\"\r\n      then have \"f x \u2208 v\" by simp\r\n      with \u2039y = f x\u203a have \"y \u2208 v\" by simp\r\n      then show \"y \u2208 f ` s \u222a v\" by simp\r\n    qed\r\n  qed\r\nqed\r\n\r\n(* 3\u00aa demostraci\u00f3n *)\r\n\r\nlemma \"f ` (s \u222a f -` v) \u2286 f ` s \u222a v\"\r\nproof\r\n  fix y\r\n  assume \"y \u2208 f ` (s \u222a f -` v)\"\r\n  then show \"y \u2208 f ` s \u222a v\"\r\n  proof\r\n    fix x\r\n    assume \"y = f x\"\r\n    assume \"x \u2208 s \u222a f -` v\"\r\n    then show \"y \u2208 f ` s \u222a v\"\r\n    proof\r\n      assume \"x \u2208 s\"\r\n      then show \"y \u2208 f ` s \u222a v\" by (simp add: \u2039y = f x\u203a)\r\n    next\r\n      assume \"x \u2208 f -` v\"\r\n      then show \"y \u2208 f ` s \u222a v\" by (simp add: \u2039y = f x\u203a)\r\n    qed\r\n  qed\r\nqed\r\n\r\n(* 4\u00aa demostraci\u00f3n *)\r\n\r\nlemma \"f ` (s \u222a f -` v) \u2286 f ` s \u222a v\"\r\nproof\r\n  fix y\r\n  assume \"y \u2208 f ` (s \u222a f -` v)\"\r\n  then show \"y \u2208 f ` s \u222a v\"\r\n  proof\r\n    fix x\r\n    assume \"y = f x\"\r\n    assume \"x \u2208 s \u222a f -` v\"\r\n    then show \"y \u2208 f ` s \u222a v\" using \u2039y = f x\u203a by blast\r\n  qed\r\nqed\r\n\r\n(* 5\u00aa demostraci\u00f3n *)\r\n\r\nlemma \"f ` (s \u222a f -` u) \u2286 f ` s \u222a u\"\r\n  by auto\r\n\r\nend\r\n<\/pre>\n<p>En los comentarios se pueden escribir otras soluciones, escribiendo el c\u00f3digo entre una l\u00ednea con &#60;pre lang=&quot;isar&quot;&#62; y otra con &#60;\/pre&#62;<br \/>\n[\/expand]<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar que f[s \u222a f\u207b\u00b9[v]] \u2286 f[s] \u222a v 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 variable s : set \u03b1 variable v : set \u03b2 example : f \u00bb (s \u222a f \u207b\u00b9&#8217; v) \u2286 f \u00bb s \u222a v := 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 variable s : set \u03b1 variable v : set \u03b2 &#8212; 1\u00aa demostraci\u00f3n &#8212; =============== example : f \u00bb (s \u222a f \u207b\u00b9&#8217; v) \u2286 f \u00bb s \u222a v := begin intros y hy, cases hy with x&#8230;<\/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\/485"}],"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=485"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/485\/revisions"}],"predecessor-version":[{"id":486,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/485\/revisions\/486"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=485"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=485"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=485"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}