{"id":497,"date":"2021-06-24T06:00:54","date_gmt":"2021-06-24T04:00:54","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=497"},"modified":"2021-06-15T18:28:34","modified_gmt":"2021-06-15T16:28:34","slug":"imagen-de-la-interseccion-general","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/imagen-de-la-interseccion-general\/","title":{"rendered":"Imagen de la intersecci\u00f3n general"},"content":{"rendered":"<p>Demostrar que<\/p>\n<pre lang=\"text\">\n   f[\u22c2 i, A i] \u2286 \u22c2 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 '' (\u22c2 i, A i) \u2286 \u22c2 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 '' (\u22c2 i, A i) \u2286 \u22c2 i, f '' A i :=\r\nbegin\r\n  intros y h,\r\n  apply mem_Inter_of_mem,\r\n  intro i,\r\n  cases h with x hx,\r\n  cases hx with xIA fxy,\r\n  rw \u2190 fxy,\r\n  apply mem_image_of_mem,\r\n  exact mem_Inter.mp xIA i,\r\nend\r\n\r\n-- 2\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample : f '' (\u22c2 i, A i) \u2286 \u22c2 i, f '' A i :=\r\nbegin\r\n  intros y h,\r\n  apply mem_Inter_of_mem,\r\n  intro i,\r\n  rcases h with \u27e8x, xIA, rfl\u27e9,\r\n  exact mem_image_of_mem f (mem_Inter.mp xIA i),\r\nend\r\n\r\n-- 3\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample : f '' (\u22c2 i, A i) \u2286 \u22c2 i, f '' A i :=\r\nbegin\r\n  intro y,\r\n  simp,\r\n  intros x xIA fxy i,\r\n  use [x, xIA i, fxy],\r\nend\r\n\r\n-- 4\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample : f '' (\u22c2 i, A i) \u2286 \u22c2 i, f '' A i :=\r\nby tidy\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_interseccion_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_interseccion_general\r\nimports Main\r\nbegin\r\n\r\n(* 1\u00aa demostraci\u00f3n *)\r\n\r\nlemma \"f ` (\u22c2 i \u2208 I. A i) \u2286 (\u22c2 i \u2208 I. f ` A i)\"\r\nproof (rule subsetI)\r\n  fix y\r\n  assume \"y \u2208 f ` (\u22c2 i \u2208 I. A i)\"\r\n  then show \"y \u2208 (\u22c2 i \u2208 I. f ` A i)\"\r\n  proof (rule imageE)\r\n    fix x\r\n    assume \"y = f x\"\r\n    assume xIA : \"x \u2208 (\u22c2 i \u2208 I. A i)\"\r\n    have \"f x \u2208 (\u22c2 i \u2208 I. f ` A i)\"\r\n    proof (rule INT_I)\r\n      fix i\r\n      assume \"i \u2208 I\"\r\n      with xIA have \"x \u2208 A i\"\r\n        by (rule INT_D)\r\n      then show \"f x \u2208 f ` A i\"\r\n        by (rule imageI)\r\n    qed\r\n    with \u2039y = f x\u203a show \"y \u2208 (\u22c2 i \u2208 I. f ` A i)\"\r\n      by (rule ssubst)\r\n  qed\r\nqed\r\n\r\n(* 2\u00aa demostraci\u00f3n *)\r\n\r\nlemma \"f ` (\u22c2 i \u2208 I. A i) \u2286 (\u22c2 i \u2208 I. f ` A i)\"\r\nproof\r\n  fix y\r\n  assume \"y \u2208 f ` (\u22c2 i \u2208 I. A i)\"\r\n  then show \"y \u2208 (\u22c2 i \u2208 I. f ` A i)\"\r\n  proof\r\n    fix x\r\n    assume \"y = f x\"\r\n    assume xIA : \"x \u2208 (\u22c2 i \u2208 I. A i)\"\r\n    have \"f x \u2208 (\u22c2 i \u2208 I. f ` A i)\"\r\n    proof\r\n      fix i\r\n      assume \"i \u2208 I\"\r\n      with xIA have \"x \u2208 A i\" by simp\r\n      then show \"f x \u2208 f ` A i\" by simp\r\n    qed\r\n    with \u2039y = f x\u203a show \"y \u2208 (\u22c2 i \u2208 I. f ` A i)\" by simp\r\n  qed\r\nqed\r\n\r\n(* 3\u00aa demostraci\u00f3n *)\r\n\r\nlemma \"f ` (\u22c2 i \u2208 I. A i) \u2286 (\u22c2 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[\u22c2 i, A i] \u2286 \u22c2 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 (\u22c2 i, A i) \u2286 \u22c2 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 (\u22c2 i, A i) \u2286 \u22c2 i, f \u00bb A i := begin intros y h, apply mem_Inter_of_mem,&#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\/497"}],"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=497"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/497\/revisions"}],"predecessor-version":[{"id":498,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/497\/revisions\/498"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=497"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=497"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=497"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}