{"id":678,"date":"2021-08-23T06:00:01","date_gmt":"2021-08-23T04:00:01","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=678"},"modified":"2021-08-21T16:58:44","modified_gmt":"2021-08-21T14:58:44","slug":"las-clases-de-equivalencia-de-elementos-no-relacionados-son-disjuntas","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/las-clases-de-equivalencia-de-elementos-no-relacionados-son-disjuntas\/","title":{"rendered":"Las clases de equivalencia de elementos no relacionados son disjuntas"},"content":{"rendered":"<p>Demostrar que las clases de equivalencia de elementos no relacionados son disjuntas.<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport tactic\n\nvariable  {X : Type}\nvariables {x y: X}\nvariable  {R : X \u2192 X \u2192 Prop}\n\ndef clase (R : X \u2192 X \u2192 Prop) (x : X) :=\n  {y : X | R x y}\n\nexample\n  (h : equivalence R)\n  (hxy : \u00ac R x y)\n  : clase R x \u2229 clase R y = \u2205 :=\nsorry\n<\/pre>\n<p>[expand title=\u00bbSoluciones con Lean\u00bb]<\/p>\n<pre lang=\"lean\">\r\nimport tactic\r\n\r\nvariable  {X : Type}\r\nvariables {x y: X}\r\nvariable  {R : X \u2192 X \u2192 Prop}\r\n\r\ndef clase (R : X \u2192 X \u2192 Prop) (x : X) :=\r\n  {y : X | R x y}\r\n\r\n-- 1\u00aa demostraci\u00f3n\r\nexample\r\n  (h : equivalence R)\r\n  (hxy : \u00ac R x y)\r\n  : clase R x \u2229 clase R y = \u2205 :=\r\nbegin\r\n  rcases h with \u27e8hr, hs, ht\u27e9,\r\n  by_contradiction h1,\r\n  apply hxy,\r\n  have h2 : \u2203 z, z \u2208 clase R x \u2229 clase R y,\r\n    { contrapose h1,\r\n      intro h1a,\r\n      apply h1a,\r\n      push_neg at h1,\r\n      exact set.eq_empty_iff_forall_not_mem.mpr h1, },\r\n  rcases h2 with \u27e8z, hxz, hyz\u27e9,\r\n  replace hxz : R x z := hxz,\r\n  replace hyz : R y z := hyz,\r\n  have hzy : R z y := hs hyz,\r\n  exact ht hxz hzy,\r\nend\r\n\r\n-- 2\u00aa demostraci\u00f3n\r\nexample\r\n  (h : equivalence R)\r\n  (hxy : \u00ac R x y)\r\n  : clase R x \u2229 clase R y = \u2205 :=\r\nbegin\r\n  rcases h with \u27e8hr, hs, ht\u27e9,\r\n  by_contradiction h1,\r\n  have h2 : \u2203 z, z \u2208 clase R x \u2229 clase R y,\r\n    { by finish [set.eq_empty_iff_forall_not_mem]},\r\n  apply hxy,\r\n  rcases h2 with \u27e8z, hxz, hyz\u27e9,\r\n  exact ht hxz (hs hyz),\r\nend\r\n<\/pre>\n<p>Se puede interactuar con la prueba anterior en <a href=\"https:\/\/leanprover-community.github.io\/lean-web-editor\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus\/main\/src\/Las_clases_de_equivalencia_de_elementos_no_relacionados_son_disjuntas.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;lean&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 Las_clases_de_equivalencia_de_elementos_no_relacionados_son_disjuntas\r\nimports Main\r\nbegin\r\n\r\ndefinition clase :: \"('a \u21d2 'a \u21d2 bool) \u21d2 'a \u21d2 'a set\"\r\n  where \"clase R x = {y. R x y}\"\r\n\r\n(* 1\u00aa demostraci\u00f3n *)\r\nlemma\r\n  assumes \"equivp R\"\r\n          \"\u00ac(R x y)\"\r\n  shows \"clase R x \u2229 clase R y = {}\"\r\nproof -\r\n  have \"\u2200z. z \u2208 clase R x \u27f6 z \u2209 clase R y\"\r\n  proof (intro allI impI)\r\n    fix z\r\n    assume \"z \u2208 clase R x\"\r\n    then have \"R x z\"\r\n      using clase_def by (metis CollectD)\r\n    show \"z \u2209 clase R y\"\r\n    proof (rule notI)\r\n      assume \"z \u2208 clase R y\"\r\n      then have \"R y z\"\r\n        using clase_def by (metis CollectD)\r\n      then have \"R z y\"\r\n        using assms(1) by (simp only: equivp_symp)\r\n      with \u2039R x z\u203a have \"R x y\"\r\n        using assms(1) by (simp only: equivp_transp)\r\n      with \u2039\u00acR x y\u203a show False\r\n        by (rule notE)\r\n    qed\r\n  qed\r\n  then show \"clase R x \u2229 clase R y = {}\"\r\n    by (simp only: disjoint_iff)\r\nqed\r\n\r\n(* 2\u00aa demostraci\u00f3n *)\r\nlemma\r\n  assumes \"equivp R\"\r\n          \"\u00ac(R x y)\"\r\n  shows \"clase R x \u2229 clase R y = {}\"\r\nproof -\r\n  have \"\u2200z. z \u2208 clase R x \u27f6 z \u2209 clase R y\"\r\n  proof (intro allI impI)\r\n    fix z\r\n    assume \"z \u2208 clase R x\"\r\n    then have \"R x z\"\r\n      using clase_def by fastforce\r\n    show \"z \u2209 clase R y\"\r\n    proof (rule notI)\r\n      assume \"z \u2208 clase R y\"\r\n      then have \"R y z\"\r\n        using clase_def by fastforce\r\n      then have \"R z y\"\r\n        using assms(1) by (simp only: equivp_symp)\r\n      with \u2039R x z\u203a have \"R x y\"\r\n        using assms(1) by (simp only: equivp_transp)\r\n      with \u2039\u00acR x y\u203a show False\r\n        by simp\r\n    qed\r\n  qed\r\n  then show \"clase R x \u2229 clase R y = {}\"\r\n    by (simp only: disjoint_iff)\r\nqed\r\n\r\n(* 3\u00aa demostraci\u00f3n *)\r\nlemma\r\n  assumes \"equivp R\"\r\n          \"\u00ac(R x y)\"\r\n  shows \"clase R x \u2229 clase R y = {}\"\r\n  using assms\r\n  by (metis clase_def\r\n            CollectD\r\n            equivp_symp\r\n            equivp_transp\r\n            disjoint_iff)\r\n\r\n(* 4\u00aa demostraci\u00f3n *)\r\nlemma\r\n  assumes \"equivp R\"\r\n          \"\u00ac(R x y)\"\r\n  shows \"clase R x \u2229 clase R y = {}\"\r\n  using assms\r\n  by (metis equivp_def\r\n            clase_def\r\n            CollectD\r\n            disjoint_iff_not_equal)\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 las clases de equivalencia de elementos no relacionados son disjuntas. Para ello, completar la siguiente teor\u00eda de Lean: import tactic variable {X : Type} variables {x y: X} variable {R : X \u2192 X \u2192 Prop} def clase (R : X \u2192 X \u2192 Prop) (x : X) := {y : X | R x y} example (h : equivalence R) (hxy : \u00ac R x y) : clase R x \u2229 clase R y = \u2205 := sorry [expand title=\u00bbSoluciones con Lean\u00bb] import tactic variable {X : Type} variables {x y: X} variable {R : X \u2192 X \u2192 Prop} def clase (R : X \u2192 X \u2192 Prop) (x : X) := {y : X | R x y} &#8212; 1\u00aa&#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":[30],"tags":[90,117,132,134,127,113,118,129,130,133,91,126,115,60,131,128,51,100,85,109,119,121,124,123,108,49,125,120,43,122,45,44],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/678"}],"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=678"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/678\/revisions"}],"predecessor-version":[{"id":679,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/678\/revisions\/679"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=678"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=678"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=678"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}