{"id":675,"date":"2021-08-22T06:00:36","date_gmt":"2021-08-22T04:00:36","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=675"},"modified":"2021-08-21T17:07:54","modified_gmt":"2021-08-21T15:07:54","slug":"las-clases-de-equivalencia-de-elementos-relacionados-son-iguales","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/las-clases-de-equivalencia-de-elementos-relacionados-son-iguales\/","title":{"rendered":"Las clases de equivalencia de elementos relacionados son iguales"},"content":{"rendered":"<p>Demostrar que las clases de equivalencia de elementos relacionados son iguales.<\/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 : R x y)\n  : clase R x = clase R y :=\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-- En la demostraci\u00f3n se usar\u00e1 el siguiente lema del que se presentan\r\n-- varias demostraciones.\r\n\r\n-- 1\u00aa demostraci\u00f3n del lema auxiliar\r\nexample\r\n  (h : equivalence R)\r\n  (hxy : R x y)\r\n  : clase R y \u2286 clase R x :=\r\nbegin\r\n  intros z hz,\r\n  have hyz : R y z := hz,\r\n  have htrans : transitive R := h.2.2,\r\n  have hxz : R x z := htrans hxy hyz,\r\n  exact hxz,\r\nend\r\n\r\n-- 2\u00aa demostraci\u00f3n del lema auxiliar\r\nexample\r\n  (h : equivalence R)\r\n  (hxy : R x y)\r\n  : clase R y \u2286 clase R x :=\r\nbegin\r\n  intros z hz,\r\n  exact h.2.2 hxy hz,\r\nend\r\n\r\n-- 3\u00aa demostraci\u00f3n del lema auxiliar\r\nlemma aux\r\n  (h : equivalence R)\r\n  (hxy : R x y)\r\n  : clase R y \u2286 clase R x :=\r\n\u03bb z hz, h.2.2 hxy hz\r\n\r\n-- A continuaci\u00f3n se presentan varias demostraciones del ejercicio\r\n-- usando el lema auxiliar\r\n\r\n-- 1\u00aa demostraci\u00f3n\r\nexample\r\n  (h : equivalence R)\r\n  (hxy : R x y)\r\n  : clase R x = clase R y :=\r\nbegin\r\n  apply le_antisymm,\r\n  { have hs : symmetric R := h.2.1,\r\n    have hyx : R y x := hs hxy,\r\n    exact aux h hyx, },\r\n  { exact aux h hxy, },\r\nend\r\n\r\n-- 2\u00aa demostraci\u00f3n\r\nexample\r\n  (h : equivalence R)\r\n  (hxy : R x y)\r\n  : clase R x = clase R y :=\r\nbegin\r\n  apply le_antisymm,\r\n  { exact aux h (h.2.1 hxy), },\r\n  { exact aux h hxy, },\r\nend\r\n\r\n-- 3\u00aa demostraci\u00f3n\r\nexample\r\n  (h : equivalence R)\r\n  (hxy : R x y)\r\n  : clase R x = clase R y :=\r\nle_antisymm (aux h (h.2.1 hxy)) (aux h hxy)\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_relacionados_son_iguales.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_relacionados_son_iguales\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(* En la demostraci\u00f3n se usar\u00e1 el siguiente lema del que se presentan\r\n   varias demostraciones. *)\r\n\r\n(* 1\u00aa demostraci\u00f3n del lema auxiliar *)\r\nlemma\r\n  assumes \"equivp R\"\r\n          \"R x y\"\r\n  shows \"clase R y \u2286 clase R x\"\r\nproof (rule subsetI)\r\n  fix z\r\n  assume \"z \u2208 clase R y\"\r\n  then have \"R y z\"\r\n    by (simp add: clase_def)\r\n  have \"transp R\"\r\n    using assms(1) by (rule equivp_imp_transp)\r\n  then have \"R x z\"\r\n    using \u2039R x y\u203a \u2039R y z\u203a by (rule transpD)\r\n  then show \"z \u2208 clase R x\"\r\n    by (simp add: clase_def)\r\nqed\r\n\r\n(* 2\u00aa demostraci\u00f3n del lema auxiliar *)\r\nlemma aux :\r\n  assumes \"equivp R\"\r\n          \"R x y\"\r\n  shows \"clase R y \u2286 clase R x\"\r\n  using assms\r\n  by (metis clase_def eq_refl equivp_def)\r\n\r\n(* A continuaci\u00f3n se presentan demostraciones del ejercicio *)\r\n\r\n(* 1\u00aa demostraci\u00f3n *)\r\nlemma\r\n  assumes \"equivp R\"\r\n          \"R x y\"\r\n  shows \"clase R y = clase R x\"\r\nproof (rule equalityI)\r\n  show \"clase R y \u2286 clase R x\"\r\n    using assms by (rule aux)\r\nnext\r\n  show \"clase R x \u2286 clase R y\"\r\n  proof -\r\n    have \"symp R\"\r\n      using assms(1) equivpE by blast\r\n    have \"R y x\"\r\n      using \u2039R x y\u203a by (simp add: \u2039symp R\u203a sympD)\r\n    with assms(1) show \"clase R x \u2286 clase R y\"\r\n       by (rule aux)\r\n  qed\r\nqed\r\n\r\n(* 2\u00aa demostraci\u00f3n *)\r\nlemma\r\n  assumes \"equivp R\"\r\n          \"R x y\"\r\n  shows \"clase R y = clase R x\"\r\n  using assms\r\n  by (metis clase_def equivp_def)\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 relacionados son iguales. 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 : R x y) : clase R x = clase R y := 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; En la demostraci\u00f3n se usar\u00e1&#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":[92,141,142,113,118,139,143,60,51,100,138,64,144,140,109,108,49,63,110,135,137,136],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/675"}],"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=675"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/675\/revisions"}],"predecessor-version":[{"id":676,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/675\/revisions\/676"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=675"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=675"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=675"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}