{"id":8023,"date":"2023-09-23T06:00:04","date_gmt":"2023-09-23T04:00:04","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/?p=8023"},"modified":"2023-09-26T12:40:40","modified_gmt":"2023-09-26T10:40:40","slug":"23-sep-23","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/23-sep-23\/","title":{"rendered":"La semana en Calculemus (23 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, x \u2294 y = y \u2294 x<\/a><\/li>\n<li><a href=\"#ej2\">2. En los ret\u00edculos, (x \u2293 y) \u2293 z = x \u2293 (y \u2293 z)<\/a><\/li>\n<li><a href=\"#ej3\">3. En los ret\u00edculos, (x \u2294 y) \u2294 z = x \u2294 (y \u2294 z)<\/a><\/li>\n<li><a href=\"#ej4\">4. En los ret\u00edculos, x \u2293 (x \u2294 y) = x<\/a><\/li>\n<li><a href=\"#ej5\">5. En los ret\u00edculos, x \u2294 (x \u2293 y) = x<\/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, x \u2294 y = y \u2294 x<\/h3>\n<p>Demostrar con Lean4 que en los ret\u00edculos se verifica que<br \/>\n&#92;[x \u2294 y = y \u2294 x&#92;]<br \/>\npara todo &#92;(x&#92;) e &#92;(y&#92;) en el 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 (x y z : \u03b1)\n\nexample : x \u2294 y = y \u2294 x :=\nby sorry\n<\/pre>\n<p><b>Demostraci\u00f3n en lenguaje natural<\/b><\/p>\n<p>Es consecuencia del siguiente lema auxiliar<br \/>\n&#92;[   (\u2200 a, b)[a \u2294 b \u2264 b \u2294 a] &#92;tag{1} &#92;]<br \/>\nEn efecto, sustituyendo en (1) &#92;(a&#92;) por &#92;(x&#92;) y &#92;(b&#92;) por &#92;(y&#92;), se tiene<br \/>\n&#92;[   x \u2294 y \u2264 y \u2294 x &#92;tag{2} &#92;]<br \/>\ny sustituyendo en (1) &#92;(a&#92;) por &#92;(y&#92;) y &#92;(b&#92;) por &#92;(x&#92;), se tiene<br \/>\n&#92;[   y \u2294 x \u2264 x \u2294 y &#92;tag{3} &#92;]<br \/>\nFinalmente, aplicando la propiedad antisim\u00e9trica de la divisibilidad<br \/>\na (2) y (3), se tiene<br \/>\n&#92;[   x \u2294 y = y \u2294 x &#92;]<\/p>\n<p>Para demostrar (1), por la definici\u00f3n del supremo, basta demostrar las siguientes relaciones<br \/>\n&#92;begin{align}<br \/>\n   x &amp;\u2264 y \u2294 x &#92;&#92;<br \/>\n   y &amp;\u2264 y \u2294 x<br \/>\n&#92;end{align}<br \/>\ny ambas se tienen por la definici\u00f3n del supremo.<\/p>\n<p><b>Demostraciones con Lean4<\/b><\/p>\n<pre lang=\"lean\">\nimport Mathlib.Order.Lattice\nvariable {\u03b1 : Type _} [Lattice \u03b1]\nvariable (x y z : \u03b1)\n\n-- 1\u00aa demostraci\u00f3n del lema auxiliar\nlemma aux : x \u2294 y \u2264 y \u2294 x :=\nby\n  have h1 : x \u2264 y \u2294 x :=\n    le_sup_right\n  have h2 : y \u2264 y \u2294 x :=\n    le_sup_left\n  show x \u2294 y \u2264 y \u2294 x\n  exact sup_le h1 h2\n\n-- 2\u00aa demostraci\u00f3n del lema auxiliar\nexample : x \u2294 y \u2264 y \u2294 x :=\nby\n  apply sup_le\n  { apply le_sup_right }\n  { apply le_sup_left }\n\n-- 3\u00aa demostraci\u00f3n del lema auxiliar\nexample : x \u2294 y \u2264 y \u2294 x :=\nsup_le le_sup_right le_sup_left\n\n-- 1\u00aa demostraci\u00f3n\nexample : x \u2294 y = y \u2294 x :=\nby\n  have h1 : x \u2294 y \u2264 y \u2294 x :=\n    aux x y\n  have h2 : y \u2294 x \u2264 x \u2294 y :=\n    aux y x\n  show x \u2294 y = y \u2294 x\n  exact le_antisymm h1 h2\n\n-- 2\u00aa demostraci\u00f3n\nexample : x \u2294 y = y \u2294 x :=\nby\n  apply le_antisymm\n  { apply aux }\n  { apply aux }\n\n-- 3\u00aa demostraci\u00f3n\nexample : x \u2294 y = y \u2294 x :=\nle_antisymm (aux x y) (aux y x)\n\n-- 4\u00aa demostraci\u00f3n\nexample : x \u2294 y = y \u2294 x :=\nby apply le_antisymm; simp ; simp\n\n-- 5\u00aa demostraci\u00f3n\nexample : x \u2294 y = y \u2294 x :=\n-- by apply?\nsup_comm\n\n-- Lemas usados\n-- ============\n\n-- #check (le_antisymm : x \u2264 y \u2192 y \u2264 x \u2192 x = y)\n-- #check (le_sup_left : x \u2264 x \u2294 y)\n-- #check (le_sup_right : y \u2264 x \u2294 y)\n-- #check (sup_comm : x \u2294 y = y \u2294 x)\n-- #check (sup_le : x \u2264 z \u2192 y \u2264 z \u2192 x \u2294 y \u2264 z)\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\/Conmutatividad_del_supremo.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. 20.<\/li>\n<\/ul>\n<p><a name=\"ej2\"><\/a><\/p>\n<h3>2. En los ret\u00edculos, (x \u2293 y) \u2293 z = x \u2293 (y \u2293 z)<\/h3>\n<p>Demostrar con Lean4 que en los ret\u00edculos se verifica que<br \/>\n&#92;[(x \u2293 y) \u2293 z = x \u2293 (y \u2293 z)&#92;]<\/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 (x y z : \u03b1)\n\nexample : (x \u2293 y) \u2293 z = x \u2293 (y \u2293 z) :=\nby sorry\n<\/pre>\n<p><b>Demostraci\u00f3n en lenguaje natural<\/b><\/p>\n<p>En la demostraci\u00f3n se usar\u00e1n los siguientes lemas:<br \/>\n&#92;begin{align}<br \/>\n   &amp;x \u2264 y \u2192 y \u2264 x \u2192 x = y     &#92;tag{L1} &#92;&#92;<br \/>\n   &amp;z \u2264 x \u2192 z \u2264 y \u2192 z \u2264 x \u2293 y &#92;tag{L2} &#92;&#92;<br \/>\n   &amp;x \u2293 y \u2264 x                 &#92;tag{L3} &#92;&#92;<br \/>\n   &amp;x \u2293 y \u2264 y                 &#92;tag{L4}<br \/>\n&#92;end{align}<\/p>\n<p>Por L1, es suficiente demostrar las siguientes relaciones:<br \/>\n&#92;begin{align}<br \/>\n   (x \u2293 y) \u2293 z &amp;\u2264 x \u2293 (y \u2293 z)   &#92;tag{1} &#92;&#92;<br \/>\n   x \u2293 (y \u2293 z) &amp;\u2264 (x \u2293 y) \u2293 z   &#92;tag{2}<br \/>\n&#92;end{align}<\/p>\n<p>Para demostrar (1), por L2, basta probar que<br \/>\n&#92;begin{align}<br \/>\n   (x \u2293 y) \u2293 z \u2264 x      &#92;tag{1a} &#92;&#92;<br \/>\n   (x \u2293 y) \u2293 z \u2264 y \u2293 z  &#92;tag{1b}<br \/>\n&#92;end{align}<\/p>\n<p>La (1a) se demuestra por la siguiente cadena de desigualdades<br \/>\n&#92;begin{align}<br \/>\n   (x \u2293 y) \u2293 z &amp;\u2264 x \u2293 y   &amp;&amp;&#92;text{[por L3]} &#92;&#92;<br \/>\n               &amp;\u2264 x       &amp;&amp;&#92;text{[por L3]}<br \/>\n&#92;end{align}<\/p>\n<p>Para demostrar (1b), por L2, basta probar que<br \/>\n&#92;begin{align}<br \/>\n   (x \u2293 y) \u2293 z &amp;\u2264 y &#92;tag{1b1} &#92;&#92;<br \/>\n   (x \u2293 y) \u2293 z &amp;\u2264 z &#92;tag{1b2}<br \/>\n&#92;end{align}<\/p>\n<p>La (1b1) se demuestra por la siguiente cadena de desigualdades<br \/>\n&#92;begin{align}<br \/>\n   (x \u2293 y) \u2293 z &amp;\u2264 x \u2293 y   &amp;&amp;&#92;text{[por L3]} &#92;&#92;<br \/>\n               &amp;\u2264 y       &amp;&amp;&#92;text{[por L4]}<br \/>\n&#92;end{align}<\/p>\n<p>La (1b2) se tiene por L4.<\/p>\n<p>Para demostrar (2), por L2, basta probar que<br \/>\n&#92;begin{align}<br \/>\n   x \u2293 (y \u2293 z) &amp;\u2264 x \u2293 y &#92;tag{2a} &#92;&#92;<br \/>\n   x \u2293 (y \u2293 z) &amp;\u2264 z     &#92;tag{2b}<br \/>\n&#92;end{align}<\/p>\n<p>Para demostrar (2a), por L2, basta probar que<br \/>\n&#92;begin{align}<br \/>\n   x \u2293 (y \u2293 z) &amp;\u2264 x &#92;tag{2a1} &#92;&#92;<br \/>\n   x \u2293 (y \u2293 z) &amp;\u2264 y &#92;tag{2a2}<br \/>\n&#92;end{align}<\/p>\n<p>La (2a1) se tiene por L3.<\/p>\n<p>La (2a2) se demuestra por la siguiente cadena de desigualdades<br \/>\n&#92;begin{align}<br \/>\n   x \u2293 (y \u2293 z) &amp;\u2264 y \u2293 z   &amp;&amp;&#92;text{[por L4]} &#92;&#92;<br \/>\n               &amp;\u2264 y       &amp;&amp;&#92;text{[por L3]}<br \/>\n&#92;end{align}<\/p>\n<p>La (2b) se demuestra por la siguiente cadena de desigualdades<br \/>\n&#92;begin{align}<br \/>\n   x \u2293 (y \u2293 z) &amp;\u2264 y \u2293 z   &amp;&amp;&#92;text{[por L4]} &#92;&#92;<br \/>\n               &amp;\u2264 z       &amp;&amp;&#92;text{[por L4]}<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 (x y z : \u03b1)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (x \u2293 y) \u2293 z = x \u2293 (y \u2293 z) :=\nby\n  have h1 : (x \u2293 y) \u2293 z \u2264 x \u2293 (y \u2293 z) := by\n  { have h1a : (x \u2293 y) \u2293 z \u2264 x := calc\n      (x \u2293 y) \u2293 z \u2264 x \u2293 y := by exact inf_le_left\n                _ \u2264 x     := by exact inf_le_left\n    have h1b : (x \u2293 y) \u2293 z \u2264 y \u2293 z := by\n    { have h1b1 : (x \u2293 y) \u2293 z \u2264 y := calc\n        (x \u2293 y) \u2293 z \u2264 x \u2293 y := by exact inf_le_left\n                  _ \u2264 y     := by exact inf_le_right\n      have h1b2 : (x \u2293 y) \u2293 z \u2264 z :=\n        inf_le_right\n      show (x \u2293 y) \u2293 z \u2264 y \u2293 z\n      exact le_inf h1b1 h1b2 }\n    show (x \u2293 y) \u2293 z \u2264 x \u2293 (y \u2293 z)\n    exact le_inf h1a h1b }\n  have h2 : x \u2293 (y \u2293 z) \u2264 (x \u2293 y) \u2293 z := by\n  { have h2a : x \u2293 (y \u2293 z) \u2264 x \u2293 y := by\n    { have h2a1 : x \u2293 (y \u2293 z) \u2264 x :=\n        inf_le_left\n      have h2a2 : x \u2293 (y \u2293 z) \u2264 y := calc\n        x \u2293 (y \u2293 z) \u2264 y \u2293 z := by exact inf_le_right\n                  _ \u2264 y     := by exact inf_le_left\n      show x \u2293 (y \u2293 z) \u2264 x \u2293 y\n      exact le_inf h2a1 h2a2 }\n    have h2b : x \u2293 (y \u2293 z) \u2264 z := by calc\n      x \u2293 (y \u2293 z) \u2264 y \u2293 z := by exact inf_le_right\n                _ \u2264 z     := by exact inf_le_right\n    show x \u2293 (y \u2293 z) \u2264 (x \u2293 y) \u2293 z\n    exact le_inf h2a h2b }\n  show (x \u2293 y) \u2293 z = x \u2293 (y \u2293 z)\n  exact le_antisymm h1 h2\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : x \u2293 y \u2293 z = x \u2293 (y \u2293 z) := by\n  apply le_antisymm\n  \u00b7 apply le_inf\n    \u00b7 apply le_trans\n      apply inf_le_left\n      apply inf_le_left\n    . apply le_inf\n      \u00b7 apply le_trans\n        apply inf_le_left\n        apply inf_le_right\n      . apply inf_le_right\n  . apply le_inf\n    \u00b7 apply le_inf\n      \u00b7 apply inf_le_left\n      . apply le_trans\n        apply inf_le_right\n        apply inf_le_left\n    . apply le_trans\n      apply inf_le_right\n      apply inf_le_right\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (x \u2293 y) \u2293 z = x \u2293 (y \u2293 z) :=\nby\n  apply le_antisymm\n  . apply le_inf\n    . apply inf_le_of_left_le inf_le_left\n    . apply le_inf (inf_le_of_left_le inf_le_right) inf_le_right\n  . apply le_inf\n    . apply le_inf inf_le_left (inf_le_of_right_le inf_le_left)\n    . apply inf_le_of_right_le inf_le_right\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (x \u2293 y) \u2293 z = x \u2293 (y \u2293 z) :=\nle_antisymm\n  (le_inf\n    (inf_le_of_left_le inf_le_left)\n    (le_inf (inf_le_of_left_le inf_le_right) inf_le_right))\n  (le_inf\n    (le_inf inf_le_left (inf_le_of_right_le inf_le_left))\n    (inf_le_of_right_le inf_le_right))\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (x \u2293 y) \u2293 z = x \u2293 (y \u2293 z) :=\n-- by apply?\ninf_assoc\n\n-- Lemas usados\n-- ============\n\n-- #check (inf_assoc : (x \u2293 y) \u2293 z = x \u2293 (y \u2293 z))\n-- #check (inf_le_left : x \u2293 y \u2264 x)\n-- #check (inf_le_of_left_le : x \u2264 z \u2192 x \u2293 y \u2264 z)\n-- #check (inf_le_of_right_le : y \u2264 z \u2192 x \u2293 y \u2264 z)\n-- #check (inf_le_right : x \u2293 y \u2264 y)\n-- #check (le_antisymm : x \u2264 y \u2192 y \u2264 x \u2192 x = y)\n-- #check (le_inf : z \u2264 x \u2192 z \u2264 y \u2192 z \u2264 x \u2293 y)\n-- #check (le_trans : x \u2264 y \u2192 y \u2264 z \u2192 x \u2264 z)\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\/Asociatividad_del_infimo.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. 21.<\/li>\n<\/ul>\n<p><a name=\"ej3\"><\/a><\/p>\n<h3>3. En los ret\u00edculos, (x \u2294 y) \u2294 z = x \u2294 (y \u2294 z)<\/h3>\n<p>Demostrar con Lean4 que en los ret\u00edculos se verifica que<br \/>\n&#92;[ (x \u2294 y) \u2294 z = x \u2294 (y \u2294 z) &#92;]<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Order.Lattice\n\nvariable {\u03b1 : Type _} [Lattice \u03b1]\nvariable (x y z : \u03b1)\n\nexample : (x \u2294 y) \u2294 z = x \u2294 (y \u2294 z) :=\nby sorry\n<\/pre>\n<p><b>Demostraci\u00f3n en lenguaje natural<\/b><\/p>\n<p>En la demostraci\u00f3n se usar\u00e1n los siguientes lemas<br \/>\n&#92;begin{align}<br \/>\n   &amp;x \u2264 y \u2192 y \u2264 x \u2192 x = y     &#92;tag{L1} &#92;label{L1} &#92;&#92;<br \/>\n   &amp;x \u2264 x \u2294 y                 &#92;tag{L2} &#92;label{L2} &#92;&#92;<br \/>\n   &amp;y \u2264 x \u2294 y                 &#92;tag{L3} &#92;label{L3} &#92;&#92;<br \/>\n   &amp;x \u2264 z \u2192 y \u2264 z \u2192 x \u2294 y \u2264 z &#92;tag{L4} &#92;label{L4} &#92;&#92;<br \/>\n&#92;end{align}<\/p>\n<p>Por &#92;ref{L1}, basta demostrar las siguientes relaciones:<br \/>\n&#92;begin{align}<br \/>\n   (x \u2294 y) \u2294 z &amp;\u2264 x \u2294 (y \u2294 z)  &#92;tag{1} &#92;label{1} &#92;&#92;<br \/>\n   x \u2294 (y \u2294 z) &amp;\u2264 (x \u2294 y) \u2294 z  &#92;tag{2} &#92;label{2}<br \/>\n&#92;end{align}<\/p>\n<p>Para demostrar (&#92;ref{1}), por &#92;ref{L4}, basta probar<br \/>\n&#92;begin{align}<br \/>\n   x \u2294 y &amp;\u2264 x \u2294 (y \u2294 z) &#92;tag{1a} &#92;label{1a} &#92;&#92;<br \/>\n       z &amp;\u2264 x \u2294 (y \u2294 z) &#92;tag{1b} &#92;label{1b}<br \/>\n&#92;end{align}<\/p>\n<p>Para demostrar (&#92;ref{1a}), por &#92;ref{L4}, basta probar<br \/>\n&#92;begin{align}<br \/>\n   x &amp;\u2264 x \u2294 (y \u2294 z) &#92;tag{1a1} &#92;label{1a1} &#92;&#92;<br \/>\n   y &amp;\u2264 x \u2294 (y \u2294 z) &#92;tag{1a2} &#92;label{1a2}<br \/>\n&#92;end{align}<\/p>\n<p>La (&#92;ref{1a1}) se tiene por &#92;ref{L2}.<\/p>\n<p>La (&#92;ref{1a2}) se tiene por la siguiente cadena de desigualdades:<br \/>\n&#92;begin{align}<br \/>\n   y &amp;\u2264 y \u2294 z          &amp;&amp;&#92;text{[por &#92;ref{L2}]} &#92;&#92;<br \/>\n     &amp;\u2264 x \u2294 (y \u2294 z)    &amp;&amp;&#92;text{[por &#92;ref{L3}]}<br \/>\n&#92;end{align}<\/p>\n<p>La (&#92;ref{1b}) se tiene por la siguiente cadena de desigualdades<br \/>\n&#92;begin{align}<br \/>\n   z &amp;\u2264 y \u2294 z          &amp;&amp;&#92;text{[por &#92;ref{L3}]} &#92;&#92;<br \/>\n     &amp;\u2264 x \u2294 (y \u2294 z)    &amp;&amp;&#92;text{[por &#92;ref{L3}]}<br \/>\n&#92;end{align}<\/p>\n<p>Para demostrar (&#92;ref{2}), por &#92;ref{L4}, basta probar<br \/>\n&#92;begin{align}<br \/>\n       x &amp;\u2264 (x \u2294 y) \u2294 z  &#92;tag{2a} &#92;label{2a} &#92;&#92;<br \/>\n   y \u2294 z &amp;\u2264 (x \u2294 y) \u2294 z  &#92;tag{2b} &#92;label{2b}<br \/>\n&#92;end{align}<\/p>\n<p>La (&#92;ref{2a}) se demuestra por la siguiente cadena de desigualdades:<br \/>\n&#92;begin{align}<br \/>\n   x &amp;\u2264 x \u2294 y          &amp;&amp;&#92;text{[por &#92;ref{L2}]} &#92;&#92;<br \/>\n     &amp;\u2264 (x \u2294 y) \u2294 z    &amp;&amp;&#92;text{[por &#92;ref{L2}]}<br \/>\n&#92;end{align}<\/p>\n<p>Para demostrar (&#92;ref{2b}), por &#92;ref{L4}, basta probar<br \/>\n&#92;begin{align}<br \/>\n   y &amp;\u2264 (x \u2294 y) \u2294 z &#92;tag{2b1} &#92;label{2b1} &#92;&#92;<br \/>\n   z &amp;\u2264 (x \u2294 y) \u2294 z &#92;tag{2b2} &#92;label{2b2}<br \/>\n&#92;end{align}<\/p>\n<p>La (&#92;ref{2b1}) se demuestra por la siguiente cadena de desigualdades:<br \/>\n&#92;begin{align}<br \/>\n   y &amp;\u2264 x \u2294 y          &amp;&amp;&#92;text{[por &#92;ref{L3}]} &#92;&#92;<br \/>\n     &amp;\u2264 (x \u2294 y) \u2294 z    &amp;&amp;&#92;text{[por &#92;ref{L2}]}<br \/>\n&#92;end{align}<\/p>\n<p>La (&#92;ref{2b2}) se tiene por &#92;ref{L3}.<\/p>\n<p><b>Demostraciones con Lean4<\/b><\/p>\n<pre lang=\"lean\">\nimport Mathlib.Order.Lattice\n\nvariable {\u03b1 : Type _} [Lattice \u03b1]\nvariable (x y z : \u03b1)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (x \u2294 y) \u2294 z = x \u2294 (y \u2294 z) :=\nby\n  have h1 : (x \u2294 y) \u2294 z \u2264 x \u2294 (y \u2294 z) := by\n  { have h1a : x \u2294 y \u2264 x \u2294 (y \u2294 z) := by\n    { have h1a1 : x \u2264 x \u2294 (y \u2294 z) := by exact le_sup_left\n      have h1a2 : y \u2264 x \u2294 (y \u2294 z) := calc\n        y \u2264 y \u2294 z       := by exact le_sup_left\n        _ \u2264 x \u2294 (y \u2294 z) := by exact le_sup_right\n      show x \u2294 y \u2264 x \u2294 (y \u2294 z)\n      exact sup_le h1a1 h1a2 }\n    have h1b : z \u2264 x \u2294 (y \u2294 z) := calc\n      z \u2264 y \u2294 z       := by exact le_sup_right\n      _ \u2264 x \u2294 (y \u2294 z) := by exact le_sup_right\n    show (x \u2294 y) \u2294 z \u2264 x \u2294 (y \u2294 z)\n    exact sup_le h1a h1b }\n  have h2 : x \u2294 (y \u2294 z) \u2264 (x \u2294 y) \u2294 z := by\n  { have h2a : x \u2264 (x \u2294 y) \u2294 z := calc\n      x \u2264 x \u2294 y       := by exact le_sup_left\n      _ \u2264 (x \u2294 y) \u2294 z := by exact le_sup_left\n    have h2b : y \u2294 z \u2264 (x \u2294 y) \u2294 z := by\n    { have h2b1 : y \u2264 (x \u2294 y) \u2294 z := calc\n        y \u2264 x \u2294 y       := by exact le_sup_right\n        _ \u2264 (x \u2294 y) \u2294 z := by exact le_sup_left\n      have h2b2 : z \u2264 (x \u2294 y) \u2294 z := by\n        exact le_sup_right\n      show  y \u2294 z \u2264 (x \u2294 y) \u2294 z\n      exact sup_le h2b1 h2b2 }\n    show x \u2294 (y \u2294 z) \u2264 (x \u2294 y) \u2294 z\n    exact sup_le h2a h2b }\n  show (x \u2294 y) \u2294 z = x \u2294 (y \u2294 z)\n  exact le_antisymm h1 h2\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : x \u2294 y \u2294 z = x \u2294 (y \u2294 z) :=\nby\n  apply le_antisymm\n  \u00b7 -- (x \u2294 y) \u2294 z \u2264 x \u2294 (y \u2294 z)\n    apply sup_le\n    \u00b7 -- x \u2294 y \u2264 x \u2294 (y \u2294 z)\n      apply sup_le\n      . -- x \u2264 x \u2294 (y \u2294 z)\n        apply le_sup_left\n      \u00b7 -- y \u2264 x \u2294 (y \u2294 z)\n        apply le_trans\n        . -- y \u2264 y \u2294 z\n          apply @le_sup_left _ _ y z\n        . -- y \u2294 z \u2264 x \u2294 (y \u2294 z)\n          apply le_sup_right\n    . -- z \u2264 x \u2294 (y \u2294 z)\n      apply le_trans\n      . -- z \u2264 x \u2294 (y \u2294 z)\n        apply @le_sup_right _ _ y z\n      . -- y \u2294 z \u2264 x \u2294 (y \u2294 z)\n        apply le_sup_right\n  . -- x \u2294 (y \u2294 z) \u2264 (x \u2294 y) \u2294 z\n    apply sup_le\n    \u00b7 -- x \u2264 (x \u2294 y) \u2294 z\n      apply le_trans\n      . -- x \u2264 x \u2294 y\n        apply @le_sup_left _ _ x y\n      . -- x \u2294 y \u2264 (x \u2294 y) \u2294 z\n        apply le_sup_left\n    . -- y \u2294 z \u2264 (x \u2294 y) \u2294 z\n      apply sup_le\n      \u00b7 -- y \u2264 (x \u2294 y) \u2294 z\n        apply le_trans\n        . -- y \u2264 x \u2294 y\n          apply @le_sup_right _ _ x y\n        . -- x \u2294 y \u2264 (x \u2294 y) \u2294 z\n          apply le_sup_left\n      . -- z \u2264 (x \u2294 y) \u2294 z\n        apply le_sup_right\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : x \u2294 y \u2294 z = x \u2294 (y \u2294 z) :=\nby\n  apply le_antisymm\n  \u00b7 apply sup_le\n    \u00b7 apply sup_le\n      . apply le_sup_left\n      \u00b7 apply le_trans\n        . apply @le_sup_left _ _ y z\n        . apply le_sup_right\n    . apply le_trans\n      . apply @le_sup_right _ _ y z\n      . apply le_sup_right\n  . apply sup_le\n    \u00b7 apply le_trans\n      . apply @le_sup_left _ _ x y\n      . apply le_sup_left\n    . apply sup_le\n      \u00b7 apply le_trans\n        . apply @le_sup_right _ _ x y\n        . apply le_sup_left\n      . apply le_sup_right\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (x \u2294 y) \u2294 z = x \u2294 (y \u2294 z) :=\nby\n  apply le_antisymm\n  . -- (x \u2294 y) \u2294 z \u2264 x \u2294 (y \u2294 z)\n    apply sup_le\n    . -- x \u2294 y \u2264 x \u2294 (y \u2294 z)\n      apply sup_le le_sup_left (le_sup_of_le_right le_sup_left)\n    . -- z \u2264 x \u2294 (y \u2294 z)\n      apply le_sup_of_le_right le_sup_right\n  . -- x \u2294 (y \u2294 z) \u2264 (x \u2294 y) \u2294 z\n    apply sup_le\n    . -- x \u2264 (x \u2294 y) \u2294 z\n      apply le_sup_of_le_left le_sup_left\n    . -- y \u2294 z \u2264 (x \u2294 y) \u2294 z\n      apply sup_le (le_sup_of_le_left le_sup_right) le_sup_right\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (x \u2294 y) \u2294 z = x \u2294 (y \u2294 z) :=\nby\n  apply le_antisymm\n  . apply sup_le\n    . apply sup_le le_sup_left (le_sup_of_le_right le_sup_left)\n    . apply le_sup_of_le_right le_sup_right\n  . apply sup_le\n    . apply le_sup_of_le_left le_sup_left\n    . apply sup_le (le_sup_of_le_left le_sup_right) le_sup_right\n\n-- 6\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (x \u2294 y) \u2294 z = x \u2294 (y \u2294 z) :=\nle_antisymm\n  (sup_le\n    (sup_le le_sup_left (le_sup_of_le_right le_sup_left))\n    (le_sup_of_le_right le_sup_right))\n  (sup_le\n    (le_sup_of_le_left le_sup_left)\n    (sup_le (le_sup_of_le_left le_sup_right) le_sup_right))\n\n-- 7\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (x \u2294 y) \u2294 z = x \u2294 (y \u2294 z) :=\n-- by apply?\nsup_assoc\n\n-- Lemas usados\n-- ============\n\n-- #check (le_antisymm : x \u2264 y \u2192 y \u2264 x \u2192 x = y)\n-- #check (le_sup_left : x \u2264 x \u2294 y)\n-- #check (le_sup_of_le_left : z \u2264 x \u2192 z \u2264 x \u2294 y)\n-- #check (le_sup_of_le_right : z \u2264 y \u2192 z \u2264 x \u2294 y)\n-- #check (le_sup_right : y \u2264 x \u2294 y)\n-- #check (le_trans : x \u2264 y \u2192 y \u2264 z \u2192 x \u2264 z)\n-- #check (sup_assoc : (x \u2294 y) \u2294 z = x \u2294 (y \u2294 z))\n-- #check (sup_le : x \u2264 z \u2192 y \u2264 z \u2192 x \u2294 y \u2264 z)\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\/Asociatividad_del_supremo.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. 21.<\/li>\n<\/ul>\n<p><a name=\"ej4\"><\/a><\/p>\n<h3>4. En los ret\u00edculos, x \u2293 (x \u2294 y) = x<\/h3>\n<p>Demostrar con Lean4 que en los ret\u00edculos se verifica que<br \/>\n&#92;[ x \u2293 (x \u2294 y) = x &#92;]<\/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 (x y : \u03b1)\n\nexample : x \u2293 (x \u2294 y) = x :=\nby sorry\n<\/pre>\n<p><b>Demostraci\u00f3n en lenguaje natural<\/b><\/p>\n<p>En la demostraci\u00f3n se usar\u00e1n los siguientes lemas<br \/>\n&#92;begin{align}<br \/>\n   &amp;x \u2264 y \u2192 y \u2264 x \u2192 x = y      &#92;tag{L1} &#92;&#92;<br \/>\n   &amp;x \u2293 y \u2264 x                  &#92;tag{L2} &#92;&#92;<br \/>\n   &amp;z \u2264 x \u2192 z \u2264 y \u2192 z \u2264 x \u2293 y  &#92;tag{L3} &#92;&#92;<br \/>\n   &amp;x \u2264 x                      &#92;tag{L4} &#92;&#92;<br \/>\n   &amp;x \u2264 x \u2294 y                  &#92;tag{L5} &#92;&#92;<br \/>\n&#92;end{align}<\/p>\n<p>Por L1, basta demostrar las siguientes relaciones:<br \/>\n&#92;begin{align}<br \/>\n   &amp;x \u2293 (x \u2294 y) \u2264 x           &#92;tag{1} &#92;&#92;<br \/>\n   &amp;x \u2264 x \u2293 (x \u2294 y) &#92;tag{2}<br \/>\n&#92;end{align}<\/p>\n<p>La (1) se tiene por L2.<\/p>\n<p>Para demostrar la (2), por L3, basta probar las relaciones:<br \/>\n&#92;begin{align}<br \/>\n   x &amp;\u2264 x       &#92;tag{2a} &#92;&#92;<br \/>\n   x &amp;\u2264 x \u2294 y   &#92;tag{2b}<br \/>\n&#92;end{align}<\/p>\n<p>La (2a) se tiene por L4.<\/p>\n<p>La (2b) se tiene por L5<\/p>\n<p><b>Demostraciones con Lean4<\/b><\/p>\n<pre lang=\"lean\">\nimport Mathlib.Order.Lattice\nvariable {\u03b1 : Type _} [Lattice \u03b1]\nvariable (x y : \u03b1)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : x \u2293 (x \u2294 y) = x :=\nby\n  have h1 : x \u2293 (x \u2294 y) \u2264 x := inf_le_left\n  have h2 : x \u2264 x \u2293 (x \u2294 y)\n  { have h2a : x \u2264 x := le_rfl\n    have h2b : x \u2264 x \u2294 y := le_sup_left\n    show x \u2264 x \u2293 (x \u2294 y)\n    exact le_inf h2a h2b }\n  show x \u2293 (x \u2294 y) = x\n  exact le_antisymm h1 h2\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : x \u2293 (x \u2294 y) = x :=\nby\n  have h1 : x \u2293 (x \u2294 y) \u2264 x := by simp\n  have h2 : x \u2264 x \u2293 (x \u2294 y) := by simp\n  show x \u2293 (x \u2294 y) = x\n  exact le_antisymm h1 h2\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : x \u2293 (x \u2294 y) = x :=\nby\n  apply le_antisymm\n  . -- x \u2293 (x \u2294 y) \u2264 x\n    apply inf_le_left\n  . -- x \u2264 x \u2293 (x \u2294 y)\n    apply le_inf\n    . -- x \u2264 x\n      apply le_rfl\n    . -- x \u2264 x \u2294 y\n      apply le_sup_left\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : x \u2293 (x \u2294 y) = x :=\nle_antisymm inf_le_left (le_inf le_rfl le_sup_left)\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample : x \u2293 (x \u2294 y) = x :=\n-- by apply?\ninf_sup_self\n\n-- 6\u00aa demostraci\u00f3n\n-- ===============\n\nexample : x \u2293 (x \u2294 y) = x :=\nby simp\n\n-- Lemas usados\n-- ============\n\n-- variable (z : \u03b1)\n-- #check (inf_le_left : x \u2293 y \u2264 x)\n-- #check (inf_sup_self : x \u2293 (x \u2294 y) = x)\n-- #check (le_antisymm : x \u2264 y \u2192 y \u2264 x \u2192 x = y)\n-- #check (le_inf : z \u2264 x \u2192 z \u2264 y \u2192 z \u2264 x \u2293 y)\n-- #check (le_rfl : x \u2264 x)\n-- #check (le_sup_left : x \u2264 x \u2294 y)\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\/Leyes_de_absorcion_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. 21.<\/li>\n<\/ul>\n<p><a name=\"ej5\"><\/a><\/p>\n<h3>5. En los ret\u00edculos, x \u2294 (x \u2293 y) = x<\/h3>\n<p>Demostrar con Lean4 que en los ret\u00edculos se verifica que<br \/>\n&#92;[ x \u2294 (x \u2293 y) = x &#92;]<\/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 (x y : \u03b1)\n\nexample : x \u2294 (x \u2293 y) = x :=\nby sorry\n<\/pre>\n<p><b>Demostraci\u00f3n en lenguaje natural<\/b><\/p>\n<p>En la demostraci\u00f3n se usar\u00e1n los siguientes lemas<br \/>\n&#92;begin{align}<br \/>\n  &amp;x \u2264 y \u2192 y \u2264 x \u2192 x = y       &#92;tag{L1} &#92;&#92;<br \/>\n  &amp;x \u2293 y \u2264 x                   &#92;tag{L2} &#92;&#92;<br \/>\n  &amp;x \u2264 x                       &#92;tag{L3} &#92;&#92;<br \/>\n  &amp;x \u2264 x \u2294 y                   &#92;tag{L4} &#92;&#92;<br \/>\n  &amp;x \u2264 z \u2192 y \u2264 z \u2192 x \u2294 y \u2264 z   &#92;tag{L5}<br \/>\n&#92;end{align}<\/p>\n<p>Por L1, basta demostrar las siguientes relaciones:<br \/>\n&#92;begin{align}<br \/>\n   &amp;x \u2294 (x \u2293 y) \u2264 x    &#92;tag{1} &#92;&#92;<br \/>\n   &amp;x \u2264 x \u2294 (x \u2293 y)    &amp;&amp;&#92;text{[que se tiene por L4]}<br \/>\n&#92;end{align}<\/p>\n<p>Para demostrar (1), por L5, basta probar las relaciones:<br \/>\n&#92;begin{align}<br \/>\n   &amp;x \u2264 x              &amp;&amp;&#92;text{[que se tiene por L3]} &#92;&#92;<br \/>\n   &amp;x \u2293 y \u2264 x          &amp;&amp;&#92;text{[que se tiene por L2]}<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 (x y : \u03b1)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : x \u2294 (x \u2293 y) = x :=\nby\n  have h1 : x \u2294 (x \u2293 y) \u2264 x\n  { have h1a : x \u2264 x := le_rfl\n    have h1b : x \u2293 y \u2264 x := inf_le_left\n    show x \u2294 (x \u2293 y) \u2264 x\n    exact sup_le h1a h1b }\n  have h2 : x \u2264 x \u2294 (x \u2293 y) := le_sup_left\n  show x \u2294 (x \u2293 y) = x\n  exact le_antisymm h1 h2\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : x \u2294 (x \u2293 y) = x :=\nby\n  have h1 : x \u2294 (x \u2293 y) \u2264 x := by simp\n  have h2 : x \u2264 x \u2294 (x \u2293 y) := by simp\n  show x \u2294 (x \u2293 y) = x\n  exact le_antisymm h1 h2\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : x \u2294 (x \u2293 y) = x :=\nby\n  apply le_antisymm\n  . -- x \u2294 (x \u2293 y) \u2264 x\n    apply sup_le\n    . -- x \u2264 x\n      apply le_rfl\n    . -- x \u2293 y \u2264 x\n      apply inf_le_left\n  . -- x \u2264 x \u2294 (x \u2293 y)\n    apply le_sup_left\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : x \u2294 (x \u2293 y) = x :=\n-- by apply?\nsup_inf_self\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample : x \u2294 (x \u2293 y) = x :=\nby simp\n\n-- Lemas usados\n-- ============\n\n-- variable (z : \u03b1)\n-- #check (le_rfl : x \u2264 x)\n-- #check (inf_le_left : x \u2293 y \u2264 x)\n-- #check (sup_le : x \u2264 z \u2192 y \u2264 z \u2192 x \u2294 y \u2264 z)\n-- #check (le_sup_left : x \u2264 x \u2294 y)\n-- #check (le_antisymm : x \u2264 y \u2192 y \u2264 x \u2192 x = y)\n-- #check (sup_inf_self : x \u2294 (x \u2293 y) = x)\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\/Leyes_de_absorcion_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. 21.<\/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, x \u2294 y = y \u2294 x 2. En los ret\u00edculos, (x \u2293 y) \u2293 z = x \u2293 (y \u2293 z) 3. En los ret\u00edculos, (x \u2294 y) \u2294 z = x \u2294 (y \u2294 z)&#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\/8023"}],"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=8023"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/8023\/revisions"}],"predecessor-version":[{"id":8025,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/8023\/revisions\/8025"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/media?parent=8023"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/categories?post=8023"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/tags?post=8023"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}