{"id":358,"date":"2021-05-28T06:00:56","date_gmt":"2021-05-28T04:00:56","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=358"},"modified":"2021-05-27T12:14:42","modified_gmt":"2021-05-27T10:14:42","slug":"diferencia-de-union-e-interseccion","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/diferencia-de-union-e-interseccion\/","title":{"rendered":"Diferencia de uni\u00f3n e intersecci\u00f3n"},"content":{"rendered":"<p>Demostrar que<\/p>\n<blockquote><p>\n  (s \\ t) \u222a (t \\ s) = (s \u222a t) \\ (s \u2229 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) = (s \u222a t) \\ (s \u2229 t) :=\nsorry\n<\/pre>\n<h4>Soluciones<\/h4>\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 demostraci\u00f3n\n-- ===============\n\nexample : (s \\ t) \u222a (t \\ s) = (s \u222a t) \\ (s \u2229 t) :=\nbegin\n  ext x,\n  split,\n  { rintros (\u27e8xs, xnt\u27e9 | \u27e8xt, xns\u27e9),\n    { split,\n      { left,\n        exact xs },\n      { rintros \u27e8_, xt\u27e9,\n        contradiction }},\n    { split ,\n      { right,\n        exact xt },\n      { rintros \u27e8xs, _\u27e9,\n        contradiction }}},\n  { rintros \u27e8xs | xt, nxst\u27e9,\n    { left,\n      use xs,\n      intro xt,\n      apply nxst,\n      split; assumption },\n    { right,\n      use xt,\n      intro xs,\n      apply nxst,\n      split; assumption }},\nend\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (s \\ t) \u222a (t \\ s) = (s \u222a t) \\ (s \u2229 t) :=\nbegin\n  ext x,\n  split,\n  { rintros (\u27e8xs, xnt\u27e9 | \u27e8xt, xns\u27e9),\n    { finish, },\n    { finish, }},\n  { rintros \u27e8xs | xt, nxst\u27e9,\n    { finish, },\n    { finish, }},\nend\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (s \\ t) \u222a (t \\ s) = (s \u222a t) \\ (s \u2229 t) :=\nbegin\n  ext x,\n  split,\n  { rintros (\u27e8xs, xnt\u27e9 | \u27e8xt, xns\u27e9) ; finish, },\n  { rintros \u27e8xs | xt, nxst\u27e9 ; finish, },\nend\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (s \\ t) \u222a (t \\ s) = (s \u222a t) \\ (s \u2229 t) :=\nbegin\n  ext,\n  split,\n  { finish, },\n  { finish, },\nend\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (s \\ t) \u222a (t \\ s) = (s \u222a t) \\ (s \u2229 t) :=\nbegin\n  rw ext_iff,\n  intro,\n  rw iff_def,\n  finish,\nend\n\n-- 6\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (s \\ t) \u222a (t \\ s) = (s \u222a t) \\ (s \u2229 t) :=\nby finish [ext_iff, iff_def]\n<\/pre>\n<p><strong>Soluciones con Isabelle\/HOL<\/strong><\/p>\n<pre lang=\"isar\">\ntheory Diferencia_de_union_e_interseccion\nimports Main\nbegin\n\nsection \u20391 demostraci\u00f3n\u203a\n\nlemma \"(s - t) \u222a (t - s) = (s \u222a t) - (s \u2229 t)\"\nproof (rule equalityI)\n  show \"(s - t) \u222a (t - s) \u2286 (s \u222a t) - (s \u2229 t)\"\n  proof (rule subsetI)\n    fix x\n    assume \"x \u2208 (s - t) \u222a (t - s)\"\n    then show \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n    proof (rule UnE)\n      assume \"x \u2208 s - t\"\n      then show \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n      proof (rule DiffE)\n        assume \"x \u2208 s\"\n        assume \"x \u2209 t\"\n        have \"x \u2208 s \u222a t\"\n          using \u2039x \u2208 s\u203a by (simp only: UnI1)\n        moreover\n        have \"x \u2209 s \u2229 t\"\n        proof (rule notI)\n          assume \"x \u2208 s \u2229 t\"\n          then have \"x \u2208 t\"\n            by (simp only: IntD2)\n          with \u2039x \u2209 t\u203a show False\n            by (rule notE)\n        qed\n        ultimately show \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n          by (rule DiffI)\n      qed\n    next\n      assume \"x \u2208 t - s\"\n      then show \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n      proof (rule DiffE)\n        assume \"x \u2208 t\"\n        assume \"x \u2209 s\"\n        have \"x \u2208 s \u222a t\"\n          using \u2039x \u2208 t\u203a by (simp only: UnI2)\n        moreover\n        have \"x \u2209 s \u2229 t\"\n        proof (rule notI)\n          assume \"x \u2208 s \u2229 t\"\n          then have \"x \u2208 s\"\n            by (simp only: IntD1)\n          with \u2039x \u2209 s\u203a show False\n            by (rule notE)\n        qed\n        ultimately show \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n          by (rule DiffI)\n      qed\n    qed\n  qed\nnext\n  show \"(s \u222a t) - (s \u2229 t) \u2286 (s - t) \u222a (t - s)\"\n  proof (rule subsetI)\n    fix x\n    assume \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n    then show \"x \u2208 (s - t) \u222a (t - s)\"\n    proof (rule DiffE)\n      assume \"x \u2208 s \u222a t\"\n      assume \"x \u2209 s \u2229 t\"\n      note \u2039x \u2208 s \u222a t\u203a\n      then show \"x \u2208 (s - t) \u222a (t - s)\"\n      proof (rule UnE)\n        assume \"x \u2208 s\"\n        have \"x \u2209 t\"\n        proof (rule notI)\n          assume \"x \u2208 t\"\n          with \u2039x \u2208 s\u203a have \"x \u2208 s \u2229 t\"\n            by (rule IntI)\n          with \u2039x \u2209 s \u2229 t\u203a show False\n            by (rule notE)\n        qed\n        with \u2039x \u2208 s\u203a have \"x \u2208 s - t\"\n          by (rule DiffI)\n        then show \"x \u2208 (s - t) \u222a (t - s)\"\n          by (simp only: UnI1)\n      next\n        assume \"x \u2208 t\"\n        have \"x \u2209 s\"\n        proof (rule notI)\n          assume \"x \u2208 s\"\n          then have \"x \u2208 s \u2229 t\"\n            using \u2039x \u2208 t\u203a by (rule IntI)\n          with \u2039x \u2209 s \u2229 t\u203a show False\n            by (rule notE)\n        qed\n        with \u2039x \u2208 t\u203a have \"x \u2208 t - s\"\n          by (rule DiffI)\n        then show \"x \u2208 (s - t) \u222a (t - s)\"\n          by (rule UnI2)\n      qed\n    qed\n  qed\nqed\n\nsection \u20392 demostraci\u00f3n\u203a\n\nlemma \"(s - t) \u222a (t - s) = (s \u222a t) - (s \u2229 t)\"\nproof\n  show \"(s - t) \u222a (t - s) \u2286 (s \u222a t) - (s \u2229 t)\"\n  proof\n    fix x\n    assume \"x \u2208 (s - t) \u222a (t - s)\"\n    then show \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n    proof\n      assume \"x \u2208 s - t\"\n      then show \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n      proof\n        assume \"x \u2208 s\"\n        assume \"x \u2209 t\"\n        have \"x \u2208 s \u222a t\"\n          using \u2039x \u2208 s\u203a by simp\n        moreover\n        have \"x \u2209 s \u2229 t\"\n        proof\n          assume \"x \u2208 s \u2229 t\"\n          then have \"x \u2208 t\"\n            by simp\n          with \u2039x \u2209 t\u203a show False\n            by simp\n        qed\n        ultimately show \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n          by simp\n      qed\n    next\n      assume \"x \u2208 t - s\"\n      then show \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n      proof\n        assume \"x \u2208 t\"\n        assume \"x \u2209 s\"\n        have \"x \u2208 s \u222a t\"\n          using \u2039x \u2208 t\u203a by simp\n        moreover\n        have \"x \u2209 s \u2229 t\"\n        proof\n          assume \"x \u2208 s \u2229 t\"\n          then have \"x \u2208 s\"\n            by simp\n          with \u2039x \u2209 s\u203a show False\n            by simp\n        qed\n        ultimately show \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n          by simp\n      qed\n    qed\n  qed\nnext\n  show \"(s \u222a t) - (s \u2229 t) \u2286 (s - t) \u222a (t - s)\"\n  proof\n    fix x\n    assume \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n    then show \"x \u2208 (s - t) \u222a (t - s)\"\n    proof\n      assume \"x \u2208 s \u222a t\"\n      assume \"x \u2209 s \u2229 t\"\n      note \u2039x \u2208 s \u222a t\u203a\n      then show \"x \u2208 (s - t) \u222a (t - s)\"\n      proof\n        assume \"x \u2208 s\"\n        have \"x \u2209 t\"\n        proof\n          assume \"x \u2208 t\"\n          with \u2039x \u2208 s\u203a have \"x \u2208 s \u2229 t\"\n            by simp\n          with \u2039x \u2209 s \u2229 t\u203a show False\n            by simp\n        qed\n        with \u2039x \u2208 s\u203a have \"x \u2208 s - t\"\n          by simp\n        then show \"x \u2208 (s - t) \u222a (t - s)\"\n          by simp\n      next\n        assume \"x \u2208 t\"\n        have \"x \u2209 s\"\n        proof\n          assume \"x \u2208 s\"\n          then have \"x \u2208 s \u2229 t\"\n            using \u2039x \u2208 t\u203a by simp\n          with \u2039x \u2209 s \u2229 t\u203a show False\n            by simp\n        qed\n        with \u2039x \u2208 t\u203a have \"x \u2208 t - s\"\n          by simp\n        then show \"x \u2208 (s - t) \u222a (t - s)\"\n          by simp\n      qed\n    qed\n  qed\nqed\n\nsection \u20393\u00aa demostraci\u00f3n\u203a\n\nlemma \"(s - t) \u222a (t - s) = (s \u222a t) - (s \u2229 t)\"\nproof\n  show \"(s - t) \u222a (t - s) \u2286 (s \u222a t) - (s \u2229 t)\"\n  proof\n    fix x\n    assume \"x \u2208 (s - t) \u222a (t - s)\"\n    then show \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n    proof\n      assume \"x \u2208 s - t\"\n      then show \"x \u2208 (s \u222a t) - (s \u2229 t)\" by simp\n    next\n      assume \"x \u2208 t - s\"\n      then show \"x \u2208 (s \u222a t) - (s \u2229 t)\" by simp\n    qed\n  qed\nnext\n  show \"(s \u222a t) - (s \u2229 t) \u2286 (s - t) \u222a (t - s)\"\n  proof\n    fix x\n    assume \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n    then show \"x \u2208 (s - t) \u222a (t - s)\"\n    proof\n      assume \"x \u2208 s \u222a t\"\n      assume \"x \u2209 s \u2229 t\"\n      note \u2039x \u2208 s \u222a t\u203a\n      then show \"x \u2208 (s - t) \u222a (t - s)\"\n      proof\n        assume \"x \u2208 s\"\n        then show \"x \u2208 (s - t) \u222a (t - s)\"\n          using \u2039x \u2209 s \u2229 t\u203a by simp\n      next\n        assume \"x \u2208 t\"\n        then show \"x \u2208 (s - t) \u222a (t - s)\"\n          using \u2039x \u2209 s \u2229 t\u203a by simp\n      qed\n    qed\n  qed\nqed\n\nsection \u20394\u00aa demostraci\u00f3n\u203a\n\nlemma \"(s - t) \u222a (t - s) = (s \u222a t) - (s \u2229 t)\"\nproof\n  show \"(s - t) \u222a (t - s) \u2286 (s \u222a t) - (s \u2229 t)\"\n  proof\n    fix x\n    assume \"x \u2208 (s - t) \u222a (t - s)\"\n    then show \"x \u2208 (s \u222a t) - (s \u2229 t)\" by auto\n  qed\nnext\n  show \"(s \u222a t) - (s \u2229 t) \u2286 (s - t) \u222a (t - s)\"\n  proof\n    fix x\n    assume \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n    then show \"x \u2208 (s - t) \u222a (t - s)\" by auto\n  qed\nqed\n\nsection \u20395\u00aa demostraci\u00f3n\u203a\n\nlemma \"(s - t) \u222a (t - s) = (s \u222a t) - (s \u2229 t)\"\nproof\n  show \"(s - t) \u222a (t - s) \u2286 (s \u222a t) - (s \u2229 t)\" by auto\nnext\n  show \"(s \u222a t) - (s \u2229 t) \u2286 (s - t) \u222a (t - s)\" by auto\nqed\n\nsection \u20396\u00aa demostraci\u00f3n\u203a\n\nlemma \"(s - t) \u222a (t - s) = (s \u222a t) - (s \u2229 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) = (s \u222a t) \\ (s \u2229 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) = (s \u222a t) \\ (s \u2229 t) := sorry Soluciones<\/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\/358"}],"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=358"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/358\/revisions"}],"predecessor-version":[{"id":359,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/358\/revisions\/359"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=358"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=358"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=358"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}