{"id":914,"date":"2022-04-23T10:52:30","date_gmt":"2022-04-23T08:52:30","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=914"},"modified":"2022-04-24T11:12:14","modified_gmt":"2022-04-24T09:12:14","slug":"diferencia-de-diferencia-de-conjuntos","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/diferencia-de-diferencia-de-conjuntos\/","title":{"rendered":"Diferencia de diferencia de conjuntos"},"content":{"rendered":"<p>Demostrar que <code>(s \\ t) \\ u \u2286 s \\ (t \u222a u)<\/code><\/p>\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 u : set \u03b1\n\nexample : (s \\ t) \\ u \u2286 s \\ (t \u222a u) :=\nsorry\n<\/pre>\n<p><!--more--><\/p>\n<h2>Soluciones con Lean<\/h2>\n<pre lang=\"lean\">\nimport data.set.basic\nopen set\n\nvariable {\u03b1 : Type}\nvariables s t u : set \u03b1\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (s \\ t) \\ u \u2286 s \\ (t \u222a u) :=\nbegin\n  intros x hx,\n  cases hx with hxst hxnu,\n  cases hxst with hxs hxnt,\n  split,\n  { exact hxs },\n  { dsimp,\n    by_contradiction hxtu,\n    cases hxtu with hxt hxu,\n    { apply hxnt,\n      exact hxt, },\n    { apply hxnu,\n      exact hxu, }},\nend\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (s \\ t) \\ u \u2286 s \\ (t \u222a u) :=\nbegin\n  rintros x \u27e8\u27e8hxs, hxnt\u27e9, hxnu\u27e9,\n  split,\n  { exact hxs },\n  { by_contradiction hxtu,\n    cases hxtu with hxt hxu,\n    { exact hxnt hxt, },\n    { exact hxnu hxu, }},\nend\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (s \\ t) \\ u \u2286 s \\ (t \u222a u) :=\nbegin\n  rintros x \u27e8\u27e8xs, xnt\u27e9, xnu\u27e9,\n  use xs,\n  rintros (xt | xu),\n  { contradiction, },\n  { contradiction, },\nend\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (s \\ t) \\ u \u2286 s \\ (t \u222a u) :=\nbegin\n  rintros x \u27e8\u27e8xs, xnt\u27e9, xnu\u27e9,\n  use xs,\n  rintros (xt | xu); contradiction,\nend\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (s \\ t) \\ u \u2286 s \\ (t \u222a u) :=\nbegin\n  intros x xstu,\n  simp at *,\n  finish,\nend\n\n-- 6\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (s \\ t) \\ u \u2286 s \\ (t \u222a u) :=\nbegin\n  intros x xstu,\n  finish,\nend\n\n-- 7\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (s \\ t) \\ u \u2286 s \\ (t \u222a u) :=\nby rw diff_diff\n\n-- 8\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (s \\ t) \\ u \u2286 s \\ (t \u222a u) :=\nby tidy\n<\/pre>\n<p>El c\u00f3digo de las demostraciones se encuentra en <a href=\"https:\/\/github.com\/jaalonso\/Demostrando-con-Lean\/blob\/main\/src\/Diferencia_de_diferencia_de_conjuntos.lean\">GitHub<\/a> y puede ejecutarse con el <a href=\"https:\/\/leanprover-community.github.io\/lean-web-editor\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Demostrando-con-Lean\/main\/src\/Diferencia_de_diferencia_de_conjuntos.lean\">Lean Web editor<\/a>.<\/p>\n<p>La construcci\u00f3n de las demostraciones se muestra en el siguiente v\u00eddeo<\/p>\n<p><iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/lHAC6Vx1aNQ\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<h2>Soluciones con Isabelle\/HOL<\/h2>\n<pre lang=\"isar\">\ntheory Diferencia_de_diferencia_de_conjuntos\nimports Main\nbegin\n\n(* 1\u00aa demostraci\u00f3n *)\nlemma \"(s - t) - u \u2286 s - (t \u222a u)\"\nproof (rule subsetI)\n  fix x\n  assume hx : \"x \u2208 (s - t) - u\"\n  then show \"x \u2208 s - (t \u222a u)\"\n  proof (rule DiffE)\n    assume xst : \"x \u2208 s - t\"\n    assume xnu : \"x \u2209 u\"\n    note xst\n    then show \"x \u2208 s - (t \u222a u)\"\n    proof (rule DiffE)\n      assume xs : \"x \u2208 s\"\n      assume xnt : \"x \u2209 t\"\n      have xntu : \"x \u2209 t \u222a u\"\n      proof (rule notI)\n        assume xtu : \"x \u2208 t \u222a u\"\n        then show False\n        proof (rule UnE)\n          assume xt : \"x \u2208 t\"\n          with xnt show False\n            by (rule notE)\n        next\n          assume xu : \"x \u2208 u\"\n          with xnu show False\n            by (rule notE)\n        qed\n      qed\n      show \"x \u2208 s - (t \u222a u)\"\n        using xs xntu by (rule DiffI)\n    qed\n  qed\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\nlemma \"(s - t) - u \u2286 s - (t \u222a u)\"\nproof\n  fix x\n  assume hx : \"x \u2208 (s - t) - u\"\n  then have xst : \"x \u2208 (s - t)\"\n    by simp\n  then have xs : \"x \u2208 s\"\n    by simp\n  have xnt : \"x \u2209 t\"\n    using xst by simp\n  have xnu : \"x \u2209 u\"\n    using hx by simp\n  have xntu : \"x \u2209 t \u222a u\"\n    using xnt xnu by simp\n  then show \"x \u2208 s - (t \u222a u)\"\n    using xs by simp\nqed\n\n(* 3\u00aa demostraci\u00f3n *)\nlemma \"(s - t) - u \u2286 s - (t \u222a u)\"\nproof\n  fix x\n  assume \"x \u2208 (s - t) - u\"\n  then show \"x \u2208 s - (t \u222a u)\"\n     by simp\nqed\n\n(* 4\u00aa demostraci\u00f3n *)\nlemma \"(s - t) - u \u2286 s - (t \u222a u)\"\nby auto\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar que (s \\ t) \\ u \u2286 s \\ (t \u222a u) Para ello, completar la siguiente teor\u00eda de Lean: import data.set.basic open set variable {\u03b1 : Type} variables s t u : set \u03b1 example : (s \\ t) \\ u \u2286 s \\ (t \u222a u) := 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":[7,1],"tags":[282],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/914"}],"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=914"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/914\/revisions"}],"predecessor-version":[{"id":919,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/914\/revisions\/919"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=914"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=914"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=914"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}