{"id":8027,"date":"2023-09-30T06:07:01","date_gmt":"2023-09-30T04:07:01","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/?p=8027"},"modified":"2023-10-01T16:22:35","modified_gmt":"2023-10-01T14:22:35","slug":"30-sep-23","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/30-sep-23\/","title":{"rendered":"La semana en Calculemus (30 de septiembre de 2023)"},"content":{"rendered":"\n<p>Esta semana he publicado en <a href=\"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/\">Calculemus<\/a> las demostraciones con Lean4 de las siguientes propiedades:<\/p>\n<ul>\n<li><a href=\"#ej1\">1. En los ret\u00edculos, una distributiva del \u00ednfimo implica la otra<\/a><\/li>\n<li><a href=\"#ej2\">2. En los ret\u00edculos, una distributiva del supremo implica la otra<\/a><\/li>\n<li><a href=\"#ej3\">3. En los anillos ordenados, a \u2264 b \u2192 0 \u2264 b &#8211; a<\/a><\/li>\n<li><a href=\"#ej4\">4. En los anillos ordenados, 0 \u2264 b &#8211; a \u2192 a \u2264 b<\/a><\/li>\n<li><a href=\"#ej5\">5. En los anillos ordenados, {a \u2264 b, 0 \u2264 c} \u22a2 ac \u2264 bc<\/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. En los ret\u00edculos, una distributiva del \u00ednfimo implica la otra<\/h3>\n<p>Demostrar con Lean4 que si \u03b1 es un ret\u00edculo tal que se<br \/>\n$$(\u2200 x, y, z) [x \u2293 (y \u2294 z) = (x \u2293 y) \u2294 (x \u2293 z))]$$<br \/>\nentonces<br \/>\n$$(a \u2294 b) \u2293 c = (a \u2293 c) \u2294 (b \u2293 c)$$<br \/>\npara todos los elementos de \u03b1.<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Order.Lattice\nvariable {\u03b1 : Type _} [Lattice \u03b1]\nvariable (a b c : \u03b1)\n\nexample\n  (h : \u2200 x y z : \u03b1, x \u2293 (y \u2294 z) = (x \u2293 y) \u2294 (x \u2293 z))\n  : (a \u2294 b) \u2293 c = (a \u2293 c) \u2294 (b \u2293 c) :=\nby sorry\n<\/pre>\n<p><b>Demostraci\u00f3n en lenguaje natural<\/b><\/p>\n<p>Se demuestra por la siguiente cadena de igualdades<br \/>\n&#92;begin{align}<br \/>\n   (a \u2294 b) \u2293 c &amp;= c \u2293 (a \u2294 b)          &amp;&amp;&#92;text{[por conmutatividad de \u2293]} &#92;&#92;<br \/>\n               &amp;= (c \u2293 a) \u2294 (c \u2293 b)    &amp;&amp;&#92;text{[por la hip\u00f3tesis]} &#92;&#92;<br \/>\n               &amp;= (a \u2293 c) \u2294 (c \u2293 b)    &amp;&amp;&#92;text{[por conmutatividad de \u2293]} &#92;&#92;\\<br \/>\n               &amp;= (a \u2293 c) \u2294 (b \u2293 c)    &amp;&amp;&#92;text{[por conmutatividad de \u2293]}<br \/>\n&#92;end{align}<\/p>\n<p><b>Demostraciones con Lean4<\/b><\/p>\n<pre lang=\"lean\">\nimport Mathlib.Order.Lattice\nvariable {\u03b1 : Type _} [Lattice \u03b1]\nvariable (a b c : \u03b1)\n\n-- 1\u00aa demostraci\u00f3n\nexample\n  (h : \u2200 x y z : \u03b1, x \u2293 (y \u2294 z) = (x \u2293 y) \u2294 (x \u2293 z))\n  : (a \u2294 b) \u2293 c = (a \u2293 c) \u2294 (b \u2293 c) :=\ncalc\n  (a \u2294 b) \u2293 c = c \u2293 (a \u2294 b)       := by rw [inf_comm]\n            _ = (c \u2293 a) \u2294 (c \u2293 b) := by rw [h]\n            _ = (a \u2293 c) \u2294 (c \u2293 b) := by rw [@inf_comm _ _ c a]\n            _ = (a \u2293 c) \u2294 (b \u2293 c) := by rw [@inf_comm _ _ c b]\n\n-- 2\u00aa demostraci\u00f3n\nexample\n  (h : \u2200 x y z : \u03b1, x \u2293 (y \u2294 z) = (x \u2293 y) \u2294 (x \u2293 z))\n  : (a \u2294 b) \u2293 c = (a \u2293 c) \u2294 (b \u2293 c) :=\nby simp [h, inf_comm]\n\n-- Lemas usados\n-- ============\n\n-- #check (inf_comm : a \u2293 b = b \u2293 a)\n<\/pre>\n<p><b>Demostraciones interactivas<\/b><\/p>\n<p>Se puede interactuar con las demostraciones anteriores en <a href=\"https:\/\/lean.math.hhu.de\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus2\/main\/src\/Propiedad_distributiva_1.lean\" rel=\"noopener noreferrer\" target=\"_blank\">Lean 4 Web<\/a>.<\/p>\n<p><b>Referencias<\/b><\/p>\n<ul>\n<li> J. Avigad y P. Massot. <a href=\"https:\/\/bit.ly\/3U4UjBk\">Mathematics in Lean<\/a>, p. 22.<\/li>\n<\/ul>\n<p><a name=\"ej2\"><\/a><\/p>\n<h3>2. En los ret\u00edculos, una distributiva del supremo implica la otra<\/h3>\n<p>Demostrar con Lean4 que si &#92;(R&#92;) es un ret\u00edculo tal que<br \/>\n&#92;[   (\u2200 x,&#92; y,&#92; z &#92;in R) [x \u2294 (y \u2293 z) = (x \u2294 y) \u2293 (x \u2294 z)] &#92;]<br \/>\nentonces<br \/>\n&#92;[   (a \u2293 b) \u2294 c = (a \u2294 c) \u2293 (b \u2294 c) &#92;]<br \/>\npara todos los elementos del ret\u00edculo.<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Order.Lattice\nvariable {\u03b1 : Type _} [Lattice \u03b1]\nvariable (a b c : \u03b1)\n\nexample\n  (h : \u2200 x y z : \u03b1, x \u2294 (y \u2293 z) = (x \u2294 y) \u2293 (x \u2294 z))\n  : (a \u2293 b) \u2294 c = (a \u2294 c) \u2293 (b \u2294 c) :=\nby sorry\n<\/pre>\n<p><b>Demostraci\u00f3n en lenguaje natural<\/b><\/p>\n<p>Se demuestra por la siguiente cadena de igualdades<br \/>\n&#92;begin{align}<br \/>\n   (a \u2293 b) \u2294 c &amp;= c \u2294 (a \u2293 b)          &amp;&amp;&#92;text{[por la conmutatividad de \u2294]} &#92;&#92;<br \/>\n               &amp;= (c \u2294 a) \u2293 (c \u2294 b)    &amp;&amp;&#92;text{[por la hip\u00f3tesis]} &#92;&#92;<br \/>\n               &amp;= (a \u2294 c) \u2293 (c \u2294 b)    &amp;&amp;&#92;text{[por la conmutatividad de \u2294]} &#92;&#92;<br \/>\n               &amp;= (a \u2294 c) \u2293 (b \u2294 c)    &amp;&amp;&#92;text{[por la conmutatividad de \u2294]}<br \/>\n&#92;end{align}<\/p>\n<p><b>Demostraciones con Lean4<\/b><\/p>\n<pre lang=\"lean\">\nimport Mathlib.Order.Lattice\nvariable {\u03b1 : Type _} [Lattice \u03b1]\nvariable (a b c : \u03b1)\n\n-- 1\u00aa demostraci\u00f3n\nexample\n  (h : \u2200 x y z : \u03b1, x \u2294 (y \u2293 z) = (x \u2294 y) \u2293 (x \u2294 z))\n  : (a \u2293 b) \u2294 c = (a \u2294 c) \u2293 (b \u2294 c) :=\ncalc\n  (a \u2293 b) \u2294 c = c \u2294 (a \u2293 b)       := by rw [sup_comm]\n            _ = (c \u2294 a) \u2293 (c \u2294 b) := by rw [h]\n            _ = (a \u2294 c) \u2293 (c \u2294 b) := by rw [@sup_comm _ _ c a]\n            _ = (a \u2294 c) \u2293 (b \u2294 c) := by rw [@sup_comm _ _ c b]\n\n-- 2\u00aa demostraci\u00f3n\nexample\n  (h : \u2200 x y z : \u03b1, x \u2294 (y \u2293 z) = (x \u2294 y) \u2293 (x \u2294 z))\n  : (a \u2293 b) \u2294 c = (a \u2294 c) \u2293 (b \u2294 c) :=\nby simp [h, sup_comm]\n\n-- Lemas usados\n-- ============\n\n-- #check (sup_comm : a \u2294 b = b \u2294 a)\n<\/pre>\n<p><b>Demostraciones interactivas<\/b><\/p>\n<p>Se puede interactuar con las demostraciones anteriores en <a href=\"https:\/\/lean.math.hhu.de\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus2\/main\/src\/Propiedad_distributiva_2.lean\" rel=\"noopener noreferrer\" target=\"_blank\">Lean 4 Web<\/a>.<\/p>\n<p><b>Referencias<\/b><\/p>\n<ul>\n<li> J. Avigad y P. Massot. <a href=\"https:\/\/bit.ly\/3U4UjBk\">Mathematics in Lean<\/a>, p. 22.<\/li>\n<\/ul>\n<p><a name=\"ej3\"><\/a><\/p>\n<h3>3. En los anillos ordenados, a \u2264 b \u2192 0 \u2264 b &#8211; a<\/h3>\n<p>Demostrar con Lean4 que en los anillos ordenados se verifica que<br \/>\n&#92;[ a \u2264 b \u2192 0 \u2264 b &#8211; a &#92;]<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Algebra.Order.Ring.Defs\nvariable {R : Type _} [StrictOrderedRing R]\nvariable (a b c : R)\n\nexample : a \u2264 b \u2192 0 \u2264 b - a :=\nby sorry\n<\/pre>\n<p><b>Demostraci\u00f3n en lenguaje natural<\/b><\/p>\n<p>Se usar\u00e1n los siguientes lemas:<br \/>\n&#92;begin{align}<br \/>\n   &amp;a &#8211; a = 0                     &#92;tag{L1} &#92;&#92;<br \/>\n   &amp;a \u2264 b \u2192 (\u2200 c) [a &#8211; c \u2264 b &#8211; c] &#92;tag{L2}<br \/>\n&#92;end{align}<\/p>\n<p>Supongamos que<br \/>\n&#92;[ a \u2264 b   &#92;tag{1} &#92;]<br \/>\nLa demostraci\u00f3n se tiene por la siguiente cadena de desigualdades:<br \/>\n&#92;begin{align}<br \/>\n   0 &amp;= a &#8211; a    &amp;&amp;&#92;text{[por L1]} &#92;&#92;<br \/>\n     &amp;\u2264 b &#8211; a    &amp;&amp;&#92;text{[por (1) y L2]}<br \/>\n&#92;end{align}<\/p>\n<p><b>Demostraciones con Lean4<\/b><\/p>\n<pre lang=\"lean\">\nimport Mathlib.Algebra.Order.Ring.Defs\nvariable {R : Type _} [StrictOrderedRing R]\nvariable (a b c : R)\n\n-- 1\u00aa demostraci\u00f3n\nexample : a \u2264 b \u2192 0 \u2264 b - a :=\nby\n  intro h\n  calc\n    0 = a - a := (sub_self a).symm\n    _ \u2264 b - a := sub_le_sub_right h a\n\n-- 2\u00aa demostraci\u00f3n\nexample : a \u2264 b \u2192 0 \u2264 b - a :=\nsub_nonneg.mpr\n\n-- 3\u00aa demostraci\u00f3n\nexample : a \u2264 b \u2192 0 \u2264 b - a :=\nby simp\n\n-- Lemas usados\n-- ============\n\n-- #check (sub_le_sub_right : a \u2264 b \u2192 \u2200 (c : R), a - c \u2264 b - c)\n-- #check (sub_nonneg : 0 \u2264 a - b \u2194 b \u2264 a)\n-- #check (sub_self a : a - a = 0)\n<\/pre>\n<p><b>Demostraciones interactivas<\/b><\/p>\n<p>Se puede interactuar con las demostraciones anteriores en <a href=\"https:\/\/lean.math.hhu.de\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus2\/main\/src\/Ejercicio_sobre_anillos_ordenados_1.lean\" rel=\"noopener noreferrer\" target=\"_blank\">Lean 4 Web<\/a>.<\/p>\n<p><b>Referencias<\/b><\/p>\n<ul>\n<li> J. Avigad y P. Massot. <a href=\"https:\/\/bit.ly\/3U4UjBk\">Mathematics in Lean<\/a>, p. 22.<\/li>\n<\/ul>\n<p><a name=\"ej4\"><\/a><\/p>\n<h3>4. En los anillos ordenados, 0 \u2264 b &#8211; a \u2192 a \u2264 b<\/h3>\n<p>Demostrar con Lean4 que en los anillos ordenados<br \/>\n&#92;[ 0 \u2264 b &#8211; a \u2192 a \u2264 b &#92;]<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Algebra.Order.Ring.Defs\nvariable {R : Type _} [StrictOrderedRing R]\nvariable (a b c : R)\n\nexample : 0 \u2264 b - a \u2192 a \u2264 b :=\nby sorry\n<\/pre>\n<p><b>Demostraci\u00f3n en lenguaje natural<\/b><\/p>\n<p>Se usar\u00e1n los siguientes lemas:<br \/>\n&#92;begin{align}<br \/>\n   &amp;0 + a = a                     &#92;tag{L1} &#92;&#92;<br \/>\n   &amp;b \u2264 c \u2192 (\u2200 a) [b + a \u2264 c + a] &#92;tag{L2} &#92;&#92;<br \/>\n   &amp;a &#8211; b + b = -a                &#92;tag{L3}<br \/>\n&#92;end{align}<br \/>\nSupongamos que<br \/>\n&#92;[ 0 \u2264 b &#8211; a  &#92;tag{1} &#92;]<br \/>\nLa demostraci\u00f3n se tiene por la siguiente cadena de desigualdades:<br \/>\n&#92;begin{align}<br \/>\n   a &amp;= 0 + a          &amp;&amp;&#92;text{[por L1]} &#92;&#92;<br \/>\n     &amp;\u2264 (b &#8211; a) + a    &amp;&amp;&#92;text{[por (1) y L2]} &#92;&#92;<br \/>\n     &amp;= b              &amp;&amp;&#92;text{[por L3]}<br \/>\n&#92;end{align}<\/p>\n<p><b>Demostraciones con Lean4<\/b><\/p>\n<pre lang=\"lean\">\nimport Mathlib.Algebra.Order.Ring.Defs\nvariable {R : Type _} [StrictOrderedRing R]\nvariable (a b c : R)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : 0 \u2264 b - a \u2192 a \u2264 b :=\nby\n  intro h\n  calc\n    a = 0 + a       := (zero_add a).symm\n    _ \u2264 (b - a) + a := add_le_add_right h a\n    _ = b           := sub_add_cancel b a\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : 0 \u2264 b - a \u2192 a \u2264 b :=\n-- by apply?\nsub_nonneg.mp\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : 0 \u2264 b - a \u2192 a \u2264 b :=\nby simp\n\n-- Lemas usados\n-- ============\n\n-- #check (zero_add a : 0 + a = a)\n-- #check (add_le_add_right : b \u2264 c \u2192 \u2200 (a : R),  b + a \u2264 c + a)\n-- #check (sub_add_cancel a b : a - b + b = a)\n-- #check (sub_nonneg : 0 \u2264 a - b \u2194 b \u2264 a)\n<\/pre>\n<p><b>Demostraciones interactivas<\/b><\/p>\n<p>Se puede interactuar con las demostraciones anteriores en <a href=\"https:\/\/lean.math.hhu.de\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus2\/main\/src\/Ejercicio_sobre_anillos_ordenados_2.lean\" rel=\"noopener noreferrer\" target=\"_blank\">Lean 4 Web<\/a>.<\/p>\n<p><b>Referencias<\/b><\/p>\n<ul>\n<li> J. Avigad y P. Massot. <a href=\"https:\/\/bit.ly\/3U4UjBk\">Mathematics in Lean<\/a>, p. 22.<\/li>\n<\/ul>\n<p><a name=\"ej5\"><\/a><\/p>\n<h3>5. En los anillos ordenados, {a \u2264 b, 0 \u2264 c} \u22a2 ac \u2264 bc<\/h3>\n<p>Demostrar con Lean4 que, en los anillos ordenados,<br \/>\n&#92;[ &#92;{a \u2264 b, 0 \u2264 c&#92;} \u22a2 ac \u2264 bc &#92;]<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Algebra.Order.Ring.Defs\nvariable {R : Type _} [StrictOrderedRing R]\nvariable (a b c : R)\n\nexample\n  (h1 : a \u2264 b)\n  (h2 : 0 \u2264 c)\n  : a * c \u2264 b * c :=\nby sorry\n<\/pre>\n<p><b>Demostraci\u00f3n en lenguaje natural<\/b><\/p>\n<p>Se usar\u00e1n los siguientes lemas:<br \/>\n&#92;begin{align}<br \/>\n   &amp;0 \u2264 a &#8211; b \u2194 b \u2264 a           &#92;tag{L1} &#92;&#92;<br \/>\n   &amp;0 \u2264 a \u2192 0 \u2264 b \u2192 0 \u2264 ab      &#92;tag{L2} &#92;&#92;<br \/>\n   &amp;(a &#8211; b)c = ac &#8211; bc          &#92;tag{L3}<br \/>\n&#92;end{align}<\/p>\n<p>Supongamos que<br \/>\n&#92;begin{align}<br \/>\n   a &amp;\u2264 b &#92;tag{1} &#92;&#92;<br \/>\n   0 &amp;\u2264 c<br \/>\n&#92;end{align}<br \/>\nDe (1), por L1, se tiene<br \/>\n&#92;[   0 \u2264 b &#8211; a &#92;]<br \/>\ny con (2), por L2, se tiene<br \/>\n&#92;[   0 \u2264 (b &#8211; a)c &#92;]<br \/>\nque, por L3, da<br \/>\n&#92;[   0 \u2264 bc &#8211; ac &#92;]<br \/>\ny, aplic\u00e1ndole L1, se tiene<br \/>\n&#92;[   ac \u2264 bc &#92;]<\/p>\n<p><b>Demostraciones con Lean4<\/b><\/p>\n<pre lang=\"lean\">\nimport Mathlib.Algebra.Order.Ring.Defs\nvariable {R : Type _} [StrictOrderedRing R]\nvariable (a b c : R)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h1 : a \u2264 b)\n  (h2 : 0 \u2264 c)\n  : a * c \u2264 b * c :=\nby\n  have h3 : 0 \u2264 b - a :=\n    sub_nonneg.mpr h1\n  have h4 : 0 \u2264 b * c - a * c := calc\n    0 \u2264 (b - a) * c   := mul_nonneg h3 h2\n    _ = b * c - a * c := sub_mul b a c\n  show a * c \u2264 b * c\n  exact sub_nonneg.mp h4\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h1 : a \u2264 b)\n  (h2 : 0 \u2264 c)\n  : a * c \u2264 b * c :=\nby\n  have h3 : 0 \u2264 b - a := sub_nonneg.mpr h1\n  have h4 : 0 \u2264 (b - a) * c := mul_nonneg h3 h2\n  -- h4 : 0 \u2264 b * c - a * c\n  rw [sub_mul] at h4\n  -- a * c \u2264 b * c\n  exact sub_nonneg.mp h4\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h1 : a \u2264 b)\n  (h2 : 0 \u2264 c)\n  : a * c \u2264 b * c :=\nby\n  -- 0 \u2264 b * c - a * c\n  apply sub_nonneg.mp\n  -- 0 \u2264 (b - a) * c\n  rw [\u2190 sub_mul]\n  apply mul_nonneg\n  . -- 0 \u2264 b - a\n    exact sub_nonneg.mpr h1\n  . -- 0 \u2264 c\n    exact h2\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h1 : a \u2264 b)\n  (h2 : 0 \u2264 c)\n  : a * c \u2264 b * c :=\nby\n  apply sub_nonneg.mp\n  rw [\u2190 sub_mul]\n  apply mul_nonneg (sub_nonneg.mpr h1) h2\n\n-- 5\u00aa demostraci\u00f3n\nexample\n  (h1 : a \u2264 b)\n  (h2 : 0 \u2264 c)\n  : a * c \u2264 b * c :=\n-- by apply?\nmul_le_mul_of_nonneg_right h1 h2\n\n-- Lemas usados\n-- ============\n\n-- #check (mul_le_mul_of_nonneg_right : a \u2264 b \u2192 0 \u2264 c \u2192 a * c \u2264 b * c)\n-- #check (mul_nonneg : 0 \u2264 a \u2192 0 \u2264 b \u2192 0 \u2264 a * b)\n-- #check (sub_mul a b c : (a - b) * c = a * c - b * c)\n-- #check (sub_nonneg : 0 \u2264 a - b \u2194 b \u2264 a)\n<\/pre>\n<p><b>Demostraciones interactivas<\/b><\/p>\n<p>Se puede interactuar con las demostraciones anteriores en <a href=\"https:\/\/lean.math.hhu.de\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus2\/main\/src\/Ejercicio_sobre_anillos_ordenados_3.lean\" rel=\"noopener noreferrer\" target=\"_blank\">Lean 4 Web<\/a>.<\/p>\n<p><b>Referencias<\/b><\/p>\n<ul>\n<li> J. Avigad y P. Massot. <a href=\"https:\/\/bit.ly\/3U4UjBk\">Mathematics in Lean<\/a>, p. 22.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Esta semana he publicado en Calculemus las demostraciones con Lean4 de las siguientes propiedades: 1. En los ret\u00edculos, una distributiva del \u00ednfimo implica la otra 2. En los ret\u00edculos, una distributiva del supremo implica la otra 3. En los anillos ordenados, a \u2264 b \u2192 0 \u2264 b &#8211; a 4. En los anillos ordenados,&#8230;<\/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\/8027"}],"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=8027"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/8027\/revisions"}],"predecessor-version":[{"id":8028,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/8027\/revisions\/8028"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/media?parent=8027"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/categories?post=8027"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/tags?post=8027"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}