{"id":356,"date":"2021-05-27T06:00:12","date_gmt":"2021-05-27T04:00:12","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=356"},"modified":"2021-05-26T13:41:06","modified_gmt":"2021-05-26T11:41:06","slug":"union-con-su-diferencia","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/union-con-su-diferencia\/","title":{"rendered":"Uni\u00f3n con su diferencia"},"content":{"rendered":"<p>Demostrar que<\/p>\n<blockquote><p>\n  (s \\ t) \u222a t = s \u222a t\n<\/p><\/blockquote>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport data.set.basic\nopen set\n\nvariable {\u03b1 : Type}\nvariables s t : set \u03b1\n\nexample : (s \\ t) \u222a t = s \u222a t :=\nsorry\n<\/pre>\n<p><!--more--><\/p>\n<p><strong>Soluciones con Lean<\/strong><\/p>\n<pre lang=\"lean\">\nimport data.set.basic\nopen set\n\nvariable {\u03b1 : Type}\nvariables s t : set \u03b1\n\n-- 1\u00aa definici\u00f3n\n-- =============\n\nexample : (s \\ t) \u222a t = s \u222a t :=\nbegin\n  ext x,\n  split,\n  { intro hx,\n    cases hx with xst xt,\n    { left,\n      exact xst.1, },\n    { right,\n      exact xt }},\n  { by_cases h : x \u2208 t,\n    { intro _,\n      right,\n      exact h },\n    { intro hx,\n      cases hx with xs xt,\n      { left,\n        split,\n        { exact xs, },\n        { dsimp,\n          exact h, }},\n      { right,\n        exact xt, }}},\nend\n\n-- 2\u00aa definici\u00f3n\n-- =============\n\nexample : (s \\ t) \u222a t = s \u222a t :=\nbegin\n  ext x,\n  split,\n  { rintros (\u27e8xs, nxt\u27e9 | xt),\n    { left,\n      exact xs},\n    { right,\n      exact xt }},\n  { by_cases h : x \u2208 t,\n    { intro _,\n      right,\n      exact h },\n    { rintros (xs | xt),\n      { left,\n        use [xs, h] },\n      { right,\n        use xt }}},\nend\n\n-- 3\u00aa definici\u00f3n\n-- =============\n\nexample : (s \\ t) \u222a t = s \u222a t :=\nbegin\n  rw ext_iff,\n  intro,\n  rw iff_def,\n  finish,\nend\n\n-- 4\u00aa definici\u00f3n\n-- =============\n\nexample : (s \\ t) \u222a t = s \u222a t :=\nby finish [ext_iff, iff_def]\n\n-- 5\u00aa definici\u00f3n\n-- =============\n\nexample : (s \\ t) \u222a t = s \u222a t :=\ndiff_union_self\n\n-- 6\u00aa definici\u00f3n\n-- =============\n\nexample : (s \\ t) \u222a t = s \u222a t :=\nbegin\n  ext,\n  simp,\nend\n\n-- 7\u00aa definici\u00f3n\n-- =============\n\nexample : (s \\ t) \u222a t = s \u222a t :=\nby simp\n<\/pre>\n<p><strong>Soluciones con Isabelle\/HOL<\/strong><\/p>\n<pre lang=\"isar\">\ntheory Union_con_su_diferencia\nimports Main\nbegin\n\nsection \u20391\u00aa demostraci\u00f3n: Detallada\u203a\n\nlemma \"(s - t) \u222a t = s \u222a t\"\nproof (rule equalityI)\n  show \"(s - t) \u222a t \u2286 s \u222a t\"\n  proof (rule subsetI)\n    fix x\n    assume \"x \u2208 (s - t) \u222a t\"\n    then show \"x \u2208 s \u222a t\"\n    proof (rule UnE)\n      assume \"x \u2208 s - t\"\n      then have \"x \u2208 s\"\n        by (simp only: DiffD1)\n      then show \"x \u2208 s \u222a t\"\n        by (simp only: UnI1)\n    next\n      assume \"x \u2208 t\"\n      then show \"x \u2208 s \u222a t\"\n        by (simp only: UnI2)\n    qed\n  qed\nnext\n  show \"s \u222a t \u2286 (s - t) \u222a t\"\n  proof (rule subsetI)\n    fix x\n    assume \"x \u2208 s \u222a t\"\n    then show \"x \u2208 (s - t) \u222a t\"\n    proof (rule UnE)\n      assume \"x \u2208 s\"\n      show \"x \u2208 (s - t) \u222a t\"\n      proof (cases \u2039x \u2208 t\u203a)\n        assume \"x \u2208 t\"\n        then show \"x \u2208 (s - t) \u222a t\"\n          by (simp only: UnI2)\n      next\n        assume \"x \u2209 t\"\n        with \u2039x \u2208 s\u203a have \"x \u2208 s - t\"\n          by (rule DiffI)\n        then show \"x \u2208 (s - t) \u222a t\"\n          by (simp only: UnI1)\n      qed\n    next\n      assume \"x \u2208 t\"\n      then show \"x \u2208 (s - t) \u222a t\"\n        by (simp only: UnI2)\n    qed\n  qed\nqed\n\nsection \u20392\u00aa demostraci\u00f3n: Estructurada\u203a\n\nlemma \"(s - t) \u222a t = s \u222a t\"\nproof\n  show \"(s - t) \u222a t \u2286 s \u222a t\"\n  proof\n    fix x\n    assume \"x \u2208 (s - t) \u222a t\"\n    then show \"x \u2208 s \u222a t\"\n    proof\n      assume \"x \u2208 s - t\"\n      then have \"x \u2208 s\"\n        by simp\n      then show \"x \u2208 s \u222a t\"\n        by simp\n    next\n      assume \"x \u2208 t\"\n      then show \"x \u2208 s \u222a t\"\n        by simp\n    qed\n  qed\nnext\n  show \"s \u222a t \u2286 (s - t) \u222a t\"\n  proof\n    fix x\n    assume \"x \u2208 s \u222a t\"\n    then show \"x \u2208 (s - t) \u222a t\"\n    proof\n      assume \"x \u2208 s\"\n      show \"x \u2208 (s - t) \u222a t\"\n      proof\n        assume \"x \u2209 t\"\n        with \u2039x \u2208 s\u203a show \"x \u2208 s - t\"\n          by simp\n      qed\n    next\n      assume \"x \u2208 t\"\n      then show \"x \u2208 (s - t) \u222a t\"\n        by simp\n    qed\n  qed\nqed\n\nsection \u20393\u00aa demostraci\u00f3n: Con lema\u203a\n\nlemma \"(s - t) \u222a t = s \u222a t\"\nby (fact Un_Diff_cancel2)\n\nsection \u20394\u00aa demostraci\u00f3n: Autom\u00e1tica\u203a\n\nlemma \"(s - t) \u222a t = s \u222a t\"\n  by auto\n\nend\n<\/pre>\n<p><strong>Nuevas soluciones<\/strong><\/p>\n<ul>\n<li>En los comentarios se pueden escribir nuevas soluciones.\n<li>El c\u00f3digo se debe escribir entre una l\u00ednea con &#60;pre lang=&quot;isar&quot;&#62; y otra con &#60;\/pre&#62;\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar que (s \\ t) \u222a t = s \u222a t Para ello, completar la siguiente teor\u00eda de Lean: import data.set.basic open set variable {\u03b1 : Type} variables s t : set \u03b1 example : (s \\ t) \u222a t = s \u222a t := sorry<\/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\/356"}],"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=356"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/356\/revisions"}],"predecessor-version":[{"id":357,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/356\/revisions\/357"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=356"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=356"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=356"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}