{"id":494,"date":"2021-06-23T06:00:36","date_gmt":"2021-06-23T04:00:36","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=494"},"modified":"2021-06-15T13:30:24","modified_gmt":"2021-06-15T11:30:24","slug":"imagen-de-la-union-general","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/imagen-de-la-union-general\/","title":{"rendered":"Imagen de la uni\u00f3n general"},"content":{"rendered":"<p>Demostrar que<\/p>\n<pre lang=\"text\">\n   f [\u22c3 i, A i] = \u22c3 i, f [A i]\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*} {I : Type*}\nvariable  f : \u03b1 \u2192 \u03b2\nvariables A : \u2115 \u2192 set \u03b1\n\nexample : f '' (\u22c3 i, A i) = \u22c3 i, f '' A i :=\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*} {I : Type*}\r\nvariable  f : \u03b1 \u2192 \u03b2\r\nvariables A : \u2115 \u2192 set \u03b1\r\n\r\n-- 1\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample : f '' (\u22c3 i, A i) = \u22c3 i, f '' A i :=\r\nbegin\r\n  ext y,\r\n  split,\r\n  { intro hy,\r\n    rw mem_image at hy,\r\n    cases hy with x hx,\r\n    cases hx with xUA fxy,\r\n    rw mem_Union at xUA,\r\n    cases xUA with i xAi,\r\n    rw mem_Union,\r\n    use i,\r\n    rw \u2190 fxy,\r\n    apply mem_image_of_mem,\r\n    exact xAi, },\r\n  { intro hy,\r\n    rw mem_Union at hy,\r\n    cases hy with i yAi,\r\n    cases yAi with x hx,\r\n    cases hx with xAi fxy,\r\n    rw \u2190 fxy,\r\n    apply mem_image_of_mem,\r\n    rw mem_Union,\r\n    use i,\r\n    exact xAi, },\r\nend\r\n\r\n-- 2\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample : f '' (\u22c3 i, A i) = \u22c3 i, f '' A i :=\r\nbegin\r\n  ext y,\r\n  simp,\r\n  split,\r\n  { rintros \u27e8x, \u27e8i, xAi\u27e9, fxy\u27e9,\r\n    use [i, x, xAi, fxy] },\r\n  { rintros \u27e8i, x, xAi, fxy\u27e9,\r\n    exact \u27e8x, \u27e8i, xAi\u27e9, fxy\u27e9 },\r\nend\r\n\r\n-- 3\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample : f '' (\u22c3 i, A i) = \u22c3 i, f '' A i :=\r\nby tidy\r\n\r\n-- 4\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample : f '' (\u22c3 i, A i) = \u22c3 i, f '' A i :=\r\nimage_Union\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\/Imagen_de_la_union_general.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 Imagen_de_la_union_general\r\nimports Main\r\nbegin\r\n\r\n(* 1\u00aa demostraci\u00f3n *)\r\n\r\nlemma \"f ` (\u22c3 i \u2208 I. A i) = (\u22c3 i \u2208 I. f ` A i)\"\r\nproof (rule equalityI)\r\n  show \"f ` (\u22c3 i \u2208 I. A i) \u2286 (\u22c3 i \u2208 I. f ` A i)\"\r\n  proof (rule subsetI)\r\n    fix y\r\n    assume \"y \u2208 f ` (\u22c3 i \u2208 I. A i)\"\r\n    then show \"y \u2208 (\u22c3 i \u2208 I. f ` A i)\"\r\n    proof (rule imageE)\r\n      fix x\r\n      assume \"y = f x\"\r\n      assume \"x \u2208 (\u22c3 i \u2208 I. A i)\"\r\n      then have \"f x \u2208 (\u22c3 i \u2208 I. f ` A i)\"\r\n      proof (rule UN_E)\r\n        fix i\r\n        assume \"i \u2208 I\"\r\n        assume \"x \u2208 A i\"\r\n        then have \"f x \u2208 f ` A i\"\r\n          by (rule imageI)\r\n        with \u2039i \u2208 I\u203a show \"f x \u2208 (\u22c3 i \u2208 I. f ` A i)\"\r\n          by (rule UN_I)\r\n      qed\r\n      with \u2039y = f x\u203a show \"y \u2208 (\u22c3 i \u2208 I. f ` A i)\"\r\n        by (rule ssubst)\r\n    qed\r\n  qed\r\nnext\r\n  show \"(\u22c3 i \u2208 I. f ` A i) \u2286 f ` (\u22c3 i \u2208 I. A i)\"\r\n  proof (rule subsetI)\r\n    fix y\r\n    assume \"y \u2208 (\u22c3 i \u2208 I. f ` A i)\"\r\n    then show \"y \u2208 f ` (\u22c3 i \u2208 I. A i)\"\r\n    proof (rule UN_E)\r\n      fix i\r\n      assume \"i \u2208 I\"\r\n      assume \"y \u2208 f ` A i\"\r\n      then show \"y \u2208 f ` (\u22c3 i \u2208 I. A i)\"\r\n      proof (rule imageE)\r\n        fix x\r\n        assume \"y = f x\"\r\n        assume \"x \u2208 A i\"\r\n        with \u2039i \u2208 I\u203a have \"x \u2208 (\u22c3 i \u2208 I. A i)\"\r\n          by (rule UN_I)\r\n        then have \"f x \u2208 f ` (\u22c3 i \u2208 I. A i)\"\r\n          by (rule imageI)\r\n        with \u2039y = f x\u203a show \"y \u2208 f ` (\u22c3 i \u2208 I. A i)\"\r\n          by (rule ssubst)\r\n      qed\r\n    qed\r\n  qed\r\nqed\r\n\r\n(* 2\u00aa demostraci\u00f3n *)\r\n\r\nlemma \"f ` (\u22c3 i \u2208 I. A i) = (\u22c3 i \u2208 I. f ` A i)\"\r\nproof\r\n  show \"f ` (\u22c3 i \u2208 I. A i) \u2286 (\u22c3 i \u2208 I. f ` A i)\"\r\n  proof\r\n    fix y\r\n    assume \"y \u2208 f ` (\u22c3 i \u2208 I. A i)\"\r\n    then show \"y \u2208 (\u22c3 i \u2208 I. f ` A i)\"\r\n    proof\r\n      fix x\r\n      assume \"y = f x\"\r\n      assume \"x \u2208 (\u22c3 i \u2208 I. A i)\"\r\n      then have \"f x \u2208 (\u22c3 i \u2208 I. f ` A i)\"\r\n      proof\r\n        fix i\r\n        assume \"i \u2208 I\"\r\n        assume \"x \u2208 A i\"\r\n        then have \"f x \u2208 f ` A i\" by simp\r\n        with \u2039i \u2208 I\u203a show \"f x \u2208 (\u22c3 i \u2208 I. f ` A i)\" by (rule UN_I)\r\n      qed\r\n      with \u2039y = f x\u203a show \"y \u2208 (\u22c3 i \u2208 I. f ` A i)\" by simp\r\n    qed\r\n  qed\r\nnext\r\n  show \"(\u22c3 i \u2208 I. f ` A i) \u2286 f ` (\u22c3 i \u2208 I. A i)\"\r\n  proof\r\n    fix y\r\n    assume \"y \u2208 (\u22c3 i \u2208 I. f ` A i)\"\r\n    then show \"y \u2208 f ` (\u22c3 i \u2208 I. A i)\"\r\n    proof\r\n      fix i\r\n      assume \"i \u2208 I\"\r\n      assume \"y \u2208 f ` A i\"\r\n      then show \"y \u2208 f ` (\u22c3 i \u2208 I. A i)\"\r\n      proof\r\n        fix x\r\n        assume \"y = f x\"\r\n        assume \"x \u2208 A i\"\r\n        with \u2039i \u2208 I\u203a have \"x \u2208 (\u22c3 i \u2208 I. A i)\" by (rule UN_I)\r\n        then have \"f x \u2208 f ` (\u22c3 i \u2208 I. A i)\" by simp\r\n        with \u2039y = f x\u203a show \"y \u2208 f ` (\u22c3 i \u2208 I. A i)\" by simp\r\n      qed\r\n    qed\r\n  qed\r\nqed\r\n\r\n(* 3\u00aa demostraci\u00f3n *)\r\n\r\nlemma \"f ` (\u22c3 i \u2208 I. A i) = (\u22c3 i \u2208 I. f ` A i)\"\r\n  by (simp only: image_UN)\r\n\r\n(* 4\u00aa demostraci\u00f3n *)\r\n\r\nlemma \"f ` (\u22c3 i \u2208 I. A i) = (\u22c3 i \u2208 I. f ` A i)\"\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 [\u22c3 i, A i] = \u22c3 i, f [A i] Para ello, completar la siguiente teor\u00eda de Lean: import data.set.basic import tactic open set variables {\u03b1 : Type*} {\u03b2 : Type*} {I : Type*} variable f : \u03b1 \u2192 \u03b2 variables A : \u2115 \u2192 set \u03b1 example : f \u00bb (\u22c3 i, A i) = \u22c3 i, f \u00bb A i := sorry [expand title=\u00bbSoluciones con Lean\u00bb] import data.set.basic import tactic open set variables {\u03b1 : Type*} {\u03b2 : Type*} {I : Type*} variable f : \u03b1 \u2192 \u03b2 variables A : \u2115 \u2192 set \u03b1 &#8212; 1\u00aa demostraci\u00f3n &#8212; =============== example : f \u00bb (\u22c3 i, A i) = \u22c3 i, f \u00bb A i := begin ext y, split,&#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\/494"}],"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=494"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/494\/revisions"}],"predecessor-version":[{"id":495,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/494\/revisions\/495"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=494"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=494"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=494"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}