{"id":7724,"date":"2022-04-24T11:19:31","date_gmt":"2022-04-24T09:19:31","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/?p=7724"},"modified":"2022-04-24T11:21:51","modified_gmt":"2022-04-24T09:21:51","slug":"dao-la-semana-en-calculemus-del-18-al-23-de-abril","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/dao-la-semana-en-calculemus-del-18-al-23-de-abril\/","title":{"rendered":"DAO: La semana en Calculemus (del 18 al 23 de abril)"},"content":{"rendered":"<p>Esta semana he publicado en <a href=\"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/\">Calculemus<\/a> las soluciones de los siguientes problemas:<\/p>\n<ul>\n<li><a href=\"#ej1\">1. Propiedad de monoton\u00eda de la intersecci\u00f3n<\/a><\/li>\n<li><a href=\"#ej2\">2. Propiedad distributiva de la intersecci\u00f3n sobre la uni\u00f3n<\/a><\/li>\n<li><a href=\"#ej3\">3. Diferencia de diferencia de conjuntos<\/a><\/li>\n<\/ul>\n<p>A continuaci\u00f3n se muestran las soluciones.<br \/>\n<!--more--><br \/>\n<a name=\"ej1\"><\/a><\/p>\n<h3>1. Propiedad de monoton\u00eda de la intersecci\u00f3n<\/h3>\n<p>Demostrar que la intersecci\u00f3n es mon\u00f3tona por la izquierda; es decir, si <code>s \u2286 t<\/code>, entonces <code>s \u2229 u \u2286 t \u2229 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\n  (h : s \u2286 t)\n  : s \u2229 u \u2286 t \u2229 u :=\nsorry\n<\/pre>\n<p><!--more--><\/p>\n<p><strong>Soluciones con Lean<\/strong><\/p>\n<pre lang=\"lean\">\nimport data.set.basic\nimport tactic\n\nopen set\n\nvariable {\u03b1 : Type}\nvariables s t u : set \u03b1\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : s \u2286 t)\n  : s \u2229 u \u2286 t \u2229 u :=\nbegin\n  rw subset_def,\n  rw inter_def,\n  rw inter_def,\n  dsimp,\n  intros x h1,\n  cases h1 with xs xu,\n  split,\n  { rw subset_def at h,\n    apply h,\n    exact xs, },\n  { exact xu, },\nend\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : s \u2286 t)\n  : s \u2229 u \u2286 t \u2229 u :=\nbegin\n  rw [subset_def, inter_def, inter_def],\n  dsimp,\n  rintros x \u27e8xs, xu\u27e9,\n  rw subset_def at h,\n  exact \u27e8h x xs, xu\u27e9,\nend\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : s \u2286 t)\n  : s \u2229 u \u2286 t \u2229 u :=\nbegin\n  simp only [subset_def, inter_def, inter_def],\n  rintros x \u27e8xs, xu\u27e9,\n  rw subset_def at h,\n  exact \u27e8h _ xs, xu\u27e9,\nend\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : s \u2286 t)\n  : s \u2229 u \u2286 t \u2229 u :=\nbegin\n  intros x xsu,\n  exact \u27e8h xsu.1, xsu.2\u27e9,\nend\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : s \u2286 t)\n  : s \u2229 u \u2286 t \u2229 u :=\nbegin\n  rintros x \u27e8xs, xu\u27e9,\n  exact \u27e8h xs, xu\u27e9,\nend\n\n-- 6\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : s \u2286 t)\n  : s \u2229 u \u2286 t \u2229 u :=\n-- by library_search\ninter_subset_inter_left u h\n\n-- 7\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : s \u2286 t)\n  : s \u2229 u \u2286 t \u2229 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\/Propiedad_de_monotonia_de_la_interseccion.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\/Propiedad_de_monotonia_de_la_interseccion.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\/W2_gMDHRehg\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<p><strong>Soluciones con Isabelle\/HOL<\/strong><\/p>\n<pre lang=\"isar\">\ntheory Propiedad_de_monotonia_de_la_interseccion\nimports Main\nbegin\n\n(* 1\u00aa soluci\u00f3n *)\nlemma\n  assumes \"s \u2286 t\"\n  shows   \"s \u2229 u \u2286 t \u2229 u\"\nproof  (rule subsetI)\n  fix x\n  assume hx: \"x \u2208 s \u2229 u\"\n  have xs: \"x \u2208 s\"\n    using hx\n    by (simp only: IntD1)\n  then have xt: \"x \u2208 t\"\n    using assms\n    by (simp only: subset_eq)\n  have xu: \"x \u2208 u\"\n    using hx\n    by (simp only: IntD2)\n  show \"x \u2208 t \u2229 u\"\n    using xt xu\n    by (simp only: Int_iff)\nqed\n\n(* 2 soluci\u00f3n *)\nlemma\n  assumes \"s \u2286 t\"\n  shows   \"s \u2229 u \u2286 t \u2229 u\"\nproof\n  fix x\n  assume hx: \"x \u2208 s \u2229 u\"\n  have xs: \"x \u2208 s\"\n    using hx\n    by simp\n  then have xt: \"x \u2208 t\"\n    using assms\n    by auto\n  have xu: \"x \u2208 u\"\n    using hx\n    by simp\n  show \"x \u2208 t \u2229 u\"\n    using xt xu\n    by simp\nqed\n\n(* 3\u00aa soluci\u00f3n *)\nlemma\n  assumes \"s \u2286 t\"\n  shows   \"s \u2229 u \u2286 t \u2229 u\"\nusing assms\nby auto\n\n(* 4\u00aa soluci\u00f3n *)\nlemma\n  \"s \u2286 t \u27f9 s \u2229 u \u2286 t \u2229 u\"\nby auto\n\nend\n<\/pre>\n<p><a name=\"ej2\"><\/a><\/p>\n<h3>2. Propiedad distributiva de la intersecci\u00f3n sobre la uni\u00f3n<\/h3>\n<p>Demostrar que <code>s \u2229 (t \u222a u) \u2286 (s \u2229 t) \u222a (s \u2229 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 :\n  s \u2229 (t \u222a u) \u2286 (s \u2229 t) \u222a (s \u2229 u) :=\nsorry\n<\/pre>\n<p><!--more--><\/p>\n<p><strong>Soluciones con Lean<\/strong><\/p>\n<pre lang=\"lean\">\nimport data.set.basic\nimport tactic\n\nopen set\n\nvariable {\u03b1 : Type}\nvariables s t u : set \u03b1\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample :\n  s \u2229 (t \u222a u) \u2286 (s \u2229 t) \u222a (s \u2229 u) :=\nbegin\n  intros x hx,\n  cases hx with hxs hxtu,\n  cases hxtu with hxt hxu,\n  { left,\n    split,\n    { exact hxs, },\n    { exact hxt, }},\n  { right,\n    split,\n    { exact hxs, },\n    { exact hxu, }},\nend\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample :\n  s \u2229 (t \u222a u) \u2286 (s \u2229 t) \u222a (s \u2229 u) :=\nbegin\n  rintros x \u27e8hxs, hxt | hxu\u27e9,\n  { left,\n    exact \u27e8hxs, hxt\u27e9, },\n  { right,\n    exact \u27e8hxs, hxu\u27e9, },\nend\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample :\n  s \u2229 (t \u222a u) \u2286 (s \u2229 t) \u222a (s \u2229 u) :=\nbegin\n  rintros x \u27e8hxs, hxt | hxu\u27e9,\n  { exact or.inl \u27e8hxs, hxt\u27e9, },\n  { exact or.inr \u27e8hxs, hxu\u27e9, },\nend\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample :\n  s \u2229 (t \u222a u) \u2286 (s \u2229 t) \u222a (s \u2229 u) :=\nbegin\n  intros x hx,\n  by finish,\nend\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample :\n  s \u2229 (t \u222a u) \u2286 (s \u2229 t) \u222a (s \u2229 u) :=\nby rw inter_union_distrib_left\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\/Propiedad_semidistributiva_de_la_interseccion_sobre_la_union.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\/Propiedad_semidistributiva_de_la_interseccion_sobre_la_union.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\/DRKAjEeeM_8\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<p><strong>Soluciones con Isabelle\/HOL<\/strong><\/p>\n<pre lang=\"isar\">\ntheory Propiedad_semidistributiva_de_la_interseccion_sobre_la_union\nimports Main\nbegin\n\n(* 1\u00aa demostraci\u00f3n *)\nlemma \"s \u2229 (t \u222a u) \u2286 (s \u2229 t) \u222a (s \u2229 u)\"\nproof (rule subsetI)\n  fix x\n  assume hx : \"x \u2208 s \u2229 (t \u222a u)\"\n  then have xs : \"x \u2208 s\"\n    by (simp only: IntD1)\n  have xtu: \"x \u2208 t \u222a u\"\n    using hx by (simp only: IntD2)\n  then have \"x \u2208 t \u2228 x \u2208 u\"\n    by (simp only: Un_iff)\n  then show \" x \u2208 s \u2229 t \u222a s \u2229 u\"\n  proof (rule disjE)\n    assume xt : \"x \u2208 t\"\n    have xst : \"x \u2208 s \u2229 t\"\n      using xs xt by (simp only: Int_iff)\n    then show \"x \u2208 (s \u2229 t) \u222a (s \u2229 u)\"\n      by (simp only: UnI1)\n  next\n    assume xu : \"x \u2208 u\"\n    have xst : \"x \u2208 s \u2229 u\"\n      using xs xu by (simp only: Int_iff)\n    then show \"x \u2208 (s \u2229 t) \u222a (s \u2229 u)\"\n      by (simp only: UnI2)\n  qed\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\nlemma \"s \u2229 (t \u222a u) \u2286 (s \u2229 t) \u222a (s \u2229 u)\"\nproof\n  fix x\n  assume hx : \"x \u2208 s \u2229 (t \u222a u)\"\n  then have xs : \"x \u2208 s\"\n    by simp\n  have xtu: \"x \u2208 t \u222a u\"\n    using hx by simp\n  then have \"x \u2208 t \u2228 x \u2208 u\"\n    by simp\n  then show \" x \u2208 s \u2229 t \u222a s \u2229 u\"\n  proof\n    assume xt : \"x \u2208 t\"\n    have xst : \"x \u2208 s \u2229 t\"\n      using xs xt by simp\n    then show \"x \u2208 (s \u2229 t) \u222a (s \u2229 u)\"\n      by simp\n  next\n    assume xu : \"x \u2208 u\"\n    have xst : \"x \u2208 s \u2229 u\"\n      using xs xu by simp\n    then show \"x \u2208 (s \u2229 t) \u222a (s \u2229 u)\"\n      by simp\n  qed\nqed\n\n(* 3\u00aa demostraci\u00f3n *)\nlemma \"s \u2229 (t \u222a u) \u2286 (s \u2229 t) \u222a (s \u2229 u)\"\nproof (rule subsetI)\n  fix x\n  assume hx : \"x \u2208 s \u2229 (t \u222a u)\"\n  then have xs : \"x \u2208 s\"\n    by (simp only: IntD1)\n  have xtu: \"x \u2208 t \u222a u\"\n    using hx by (simp only: IntD2)\n  then show \" x \u2208 s \u2229 t \u222a s \u2229 u\"\n  proof (rule UnE)\n    assume xt : \"x \u2208 t\"\n    have xst : \"x \u2208 s \u2229 t\"\n      using xs xt by (simp only: Int_iff)\n    then show \"x \u2208 (s \u2229 t) \u222a (s \u2229 u)\"\n      by (simp only: UnI1)\n  next\n    assume xu : \"x \u2208 u\"\n    have xst : \"x \u2208 s \u2229 u\"\n      using xs xu by (simp only: Int_iff)\n    then show \"x \u2208 (s \u2229 t) \u222a (s \u2229 u)\"\n      by (simp only: UnI2)\n  qed\nqed\n\n(* 4\u00aa demostraci\u00f3n *)\nlemma \"s \u2229 (t \u222a u) \u2286 (s \u2229 t) \u222a (s \u2229 u)\"\nproof\n  fix x\n  assume hx : \"x \u2208 s \u2229 (t \u222a u)\"\n  then have xs : \"x \u2208 s\"\n    by simp\n  have xtu: \"x \u2208 t \u222a u\"\n    using hx by simp\n  then show \" x \u2208 s \u2229 t \u222a s \u2229 u\"\n  proof (rule UnE)\n    assume xt : \"x \u2208 t\"\n    have xst : \"x \u2208 s \u2229 t\"\n      using xs xt by simp\n    then show \"x \u2208 (s \u2229 t) \u222a (s \u2229 u)\"\n      by simp\n  next\n    assume xu : \"x \u2208 u\"\n    have xst : \"x \u2208 s \u2229 u\"\n      using xs xu by simp\n    then show \"x \u2208 (s \u2229 t) \u222a (s \u2229 u)\"\n      by simp\n  qed\nqed\n\n(* 5\u00aa demostraci\u00f3n *)\nlemma \"s \u2229 (t \u222a u) \u2286 (s \u2229 t) \u222a (s \u2229 u)\"\nby (simp only: Int_Un_distrib)\n\n(* 6\u00aa demostraci\u00f3n *)\nlemma \"s \u2229 (t \u222a u) \u2286 (s \u2229 t) \u222a (s \u2229 u)\"\nby auto\n\nend\n<\/pre>\n<p><a name=\"ej3\"><\/a><\/p>\n<h3>3. Diferencia de diferencia de conjuntos<\/h3>\n<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<p><strong>Soluciones con Lean<\/strong><\/p>\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<p><strong>Soluciones con Isabelle\/HOL<\/strong><\/p>\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>Esta semana he publicado en Calculemus las soluciones de los siguientes problemas: 1. Propiedad de monoton\u00eda de la intersecci\u00f3n 2. Propiedad distributiva de la intersecci\u00f3n sobre la uni\u00f3n 3. Diferencia de diferencia de conjuntos A continuaci\u00f3n se muestran las soluciones.<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","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,"footnotes":"","_jetpack_memberships_contains_paid_content":false},"categories":[335],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_likes_enabled":false,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7724"}],"collection":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/comments?post=7724"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7724\/revisions"}],"predecessor-version":[{"id":7726,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7724\/revisions\/7726"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/media?parent=7724"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/categories?post=7724"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/tags?post=7724"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}