{"id":7985,"date":"2023-08-05T10:57:11","date_gmt":"2023-08-05T08:57:11","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/?p=7985"},"modified":"2023-08-05T11:38:44","modified_gmt":"2023-08-05T09:38:44","slug":"05-ago-23","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/05-ago-23\/","title":{"rendered":"La semana en Calculemus (5 de agosto 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. Si R es un anillo y a, b \u2208 R, entonces (a + b) + -b = a<\/a><\/li>\n<li><a href=\"#ej2\">2. Si R es un anillo y a, b, c \u2208 R tales que a+b=a+c, entonces b=c<\/a><\/li>\n<li><a href=\"#ej3\">3. Si R es un anillo y a, b, c \u2208 R tales que a+b=c+b, entonces a=c<\/a><\/li>\n<li><a href=\"#ej4\">4. Si R es un anillo y a \u2208 R, entonces a.0 = 0<\/a><\/li>\n<li><a href=\"#ej5\">5. Si R es un anillo y a \u2208 R, entonces 0.a = 0<\/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. Si R es un anillo y a, b \u2208 R, entonces (a + b) + -b = a<\/h3>\n<p>En Lean4, se declara que R es un anillo mediante la expresi\u00f3n<\/p>\n<pre lang=\"text\">\n   variable {R : Type _} [Ring R]\n<\/pre>\n<p>Como consecuencia, se tiene los siguientes axiomas<\/p>\n<pre lang=\"text\">\n   add_assoc    : \u2200 a b c : R, (a + b) + c = a + (b + c)\n   add_comm     : \u2200 a b : R,   a + b = b + a\n   zero_add     : \u2200 a : R,     0 + a = a\n   add_left_neg : \u2200 a : R,     -a + a = 0\n   mul_assoc    : \u2200 a b c : R, a * b * c = a * (b * c)\n   mul_one      : \u2200 a : R,     a * 1 = a\n   one_mul      : \u2200 a : R,     1 * a = a\n   mul_add      : \u2200 a b c : R, a * (b + c) = a * b + a * c\n   add_mul      : \u2200 a b c : R, (a + b) * c = a * c + b * c\n<\/pre>\n<p><em>Demostrar que si &#92;(R&#92;) es un anillo, entonces<br \/>\n&#92;[\\forall a, b &#92;in R, (a + b) + -b = a&#92;]<\/em><\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Algebra.Ring.Defs\n\nvariable {R : Type _} [Ring R]\nvariable (a b : R)\n\nexample : (a + b) + -b = a :=\nsorry\n<\/pre>\n<p><b>Demostraci\u00f3n en lenguaje natural<\/b><\/p>\n<p>Por la siguiente cadena de igualdades<br \/>\n\\begin{align}<br \/>\n(a + b) + -b &amp;= a + (b + -b)    &amp;&amp;\\text{[por la asociativa]} &#92;&#92;<br \/>\n             &amp;= a + 0           &amp;&amp;\\text{[por suma con opuesto]} &#92;&#92;<br \/>\n             &amp;= a               &amp;&amp;\\text{[por suma con cero]}<br \/>\n\\end{align}<\/p>\n<p><b>Demostraciones con Lean4<\/b><\/p>\n<pre lang=\"lean\">\nimport Mathlib.Algebra.Ring.Defs\n\nvariable {R : Type _} [Ring R]\nvariable (a b : R)\n\n-- 1\u00aa demostraci\u00f3n\nexample : (a + b) + -b = a :=\ncalc\n  (a + b) + -b = a + (b + -b) := by rw [add_assoc]\n             _ = a + 0        := by rw [add_right_neg]\n             _ = a            := by rw [add_zero]\n\n-- 2\u00aa demostraci\u00f3n\nexample : (a + b) + -b = a :=\nby\n  rw [add_assoc]\n  rw [add_right_neg]\n  rw [add_zero]\n\n-- 3\u00aa demostraci\u00f3n\nexample : (a + b) + -b = a :=\nby rw [add_assoc, add_right_neg, add_zero]\n\n-- 4\u00aa demostraci\u00f3n\nexample : (a + b) + -b = a :=\n  add_neg_cancel_right a b\n\n-- 5\u00aa demostraci\u00f3n\nexample : (a + b) + -b = a :=\n  add_neg_cancel_right _ _\n\n-- 6\u00aa demostraci\u00f3n\nexample : (a + b) + -b = a :=\nby simp\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\/Opuesto_se_cancela_con_la_suma_por_la_derecha.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. 11.<\/li>\n<\/ul>\n<p><a name=\"ej2\"><\/a><\/p>\n<h3>2. Si R es un anillo y a, b, c \u2208 R tales que a+b=a+c, entonces b=c<\/h3>\n<p><em>Demostrar con Lean4 que si &#92;(R&#92;) es un anillo y &#92;(a, b, c &#92;in R&#92;) tales que &#92;(a + b = a + c&#92;), entonces &#92;(b = c&#92;).<\/em><\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Algebra.Ring.Defs\nimport Mathlib.Tactic\n\nvariable {R : Type _} [Ring R]\nvariable {a b c : R}\n\nexample\n  (h : a + b = a + c)\n  : b = c :=\nsorry\n<\/pre>\n<p><b>Demostraci\u00f3n en lenguaje natural (LN)<\/b><\/p>\n<p><b>1\u00aa demostraci\u00f3n en LN<\/b><\/p>\n<p>Por la siguiente cadena de igualdades<br \/>\n\\begin{align}<br \/>\nb &amp;= 0 + b           &amp;&amp;\\text{[por suma con cero]} &#92;&#92;<br \/>\n  &amp;= (-a + a) + b    &amp;&amp;\\text{[por suma con opuesto]} &#92;&#92;<br \/>\n  &amp;= -a + (a + b)    &amp;&amp;\\text{[por asociativa]} &#92;&#92;<br \/>\n  &amp;= -a + (a + c)    &amp;&amp;\\text{[por hip\u00f3tesis]} &#92;&#92;<br \/>\n  &amp;= (-a + a) + c    &amp;&amp;\\text{[por asociativa]} &#92;&#92;<br \/>\n  &amp;= 0 + c           &amp;&amp;\\text{[por suma con opuesto]} &#92;&#92;<br \/>\n  &amp;= c               &amp;&amp;\\text{[por suma con cero]}<br \/>\n\\end{align}<\/p>\n<p><b>2\u00aa demostraci\u00f3n en LN<\/b><\/p>\n<p>Por la siguiente cadena de implicaciones<br \/>\n\\begin{align}<br \/>\na + b = a + c<br \/>\n&amp;\\Longrightarrow -a + (a + b) = -a + (a + c)     &amp;&amp;\\text{[sumando -a]} &#92;&#92;<br \/>\n&amp;\\Longrightarrow  (-a + a) + b = (-a + a) + c    &amp;&amp;\\text{[por la asociativa]} &#92;&#92;<br \/>\n&amp;\\Longrightarrow  0 + b = 0 + b                  &amp;&amp;\\text{[suma con opuesto]} &#92;&#92;<br \/>\n&amp;\\Longrightarrow  b = c                          &amp;&amp;\\text{[suma con cero]}<br \/>\n\\end{align}<\/p>\n<p><b>3\u00aa demostraci\u00f3n en LN<\/b><\/p>\n<p>Por la siguiente cadena de igualdades<br \/>\n\\begin{align}<br \/>\nb &amp;= -a + (a + b)   &#92;&#92;<br \/>\n  &amp;= -a + (a + c)   &amp;&amp;\\text{[por la hip\u00f3tesis]} &#92;&#92;<br \/>\n  &amp;= c<br \/>\n\\end{align}<\/p>\n<p><b>Demostraciones con Lean4<\/b><\/p>\n<pre lang=\"lean\">\nimport Mathlib.Algebra.Ring.Defs\nimport Mathlib.Tactic\n\nvariable {R : Type _} [Ring R]\nvariable {a b c : R}\n\n-- 1\u00aa demostraci\u00f3n\nexample\n  (h : a + b = a + c)\n  : b = c :=\ncalc\n  b = 0 + b        := by rw [zero_add]\n  _ = (-a + a) + b := by rw [add_left_neg]\n  _ = -a + (a + b) := by rw [add_assoc]\n  _ = -a + (a + c) := by rw [h]\n  _ = (-a + a) + c := by rw [\u2190add_assoc]\n  _ = 0 + c        := by rw [add_left_neg]\n  _ = c            := by rw [zero_add]\n\n-- 2\u00aa demostraci\u00f3n\nexample\n  (h : a + b = a + c)\n  : b = c :=\nby\n  have h1 : -a + (a + b) = -a + (a + c) :=\n    congrArg (HAdd.hAdd (-a)) h\n  clear h\n  rw [\u2190 add_assoc] at h1\n  rw [add_left_neg] at h1\n  rw [zero_add] at h1\n  rw [\u2190 add_assoc] at h1\n  rw [add_left_neg] at h1\n  rw [zero_add] at h1\n  exact h1\n\n-- 3\u00aa demostraci\u00f3n\nexample\n  (h : a + b = a + c)\n  : b = c :=\ncalc\n  b = -a + (a + b) := by rw [neg_add_cancel_left a b]\n  _ = -a + (a + c) := by rw [h]\n  _ = c            := by rw [neg_add_cancel_left]\n\n-- 4\u00aa demostraci\u00f3n\nexample\n  (h : a + b = a + c)\n  : b = c :=\nby\n  rw [\u2190 neg_add_cancel_left a b]\n  rw [h]\n  rw [neg_add_cancel_left]\n\n-- 5\u00aa demostraci\u00f3n\nexample\n  (h : a + b = a + c)\n  : b = c :=\nby\n  rw [\u2190 neg_add_cancel_left a b, h, neg_add_cancel_left]\n\n-- 6\u00aa demostraci\u00f3n\nexample\n  (h : a + b = a + c)\n  : b = c :=\nadd_left_cancel h\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\/Cancelativa_izquierda.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. 11.<\/li>\n<\/ul>\n<p><a name=\"ej3\"><\/a><\/p>\n<h3>3. Si R es un anillo y a, b, c \u2208 R tales que a+b=c+b, entonces a=c<\/h3>\n<p><em>Demostrar con Lean4 que si &#92;(R&#92;) es un anillo y &#92;(a, b, c &#92;in R&#92;) tales que &#92;(a + b = c + b&#92;), entonces &#92;(a = c&#92;).<\/em><\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Algebra.Ring.Defs\nimport Mathlib.Tactic\n\nvariable {R : Type _} [Ring R]\nvariable {a b c : R}\n\nexample\n  (h : a + b = c + b)\n  : a = c :=\nsorry\n<\/pre>\n<p><b>Demostraciones en lenguaje natural (LN<\/b><\/p>\n<p><b>1\u00aa demostraci\u00f3n en LN<\/b><\/p>\n<p>Por la siguiente cadena de igualdades<br \/>\n\\begin{align}<br \/>\n   a &amp;= a + 0           &amp;&amp;\\text{[por suma con cero]} &#92;&#92;<br \/>\n     &amp;= a + (b + -b)    &amp;&amp;\\text{[por suma con opuesto]} &#92;&#92;<br \/>\n     &amp;= (a + b) + -b    &amp;&amp;\\text{[por asociativa]} &#92;&#92;<br \/>\n     &amp;= (c + b) + -b    &amp;&amp;\\text{[por hip\u00f3tesis]} &#92;&#92;<br \/>\n     &amp;= c + (b + -b)    &amp;&amp;\\text{[por asociativa]} &#92;&#92;<br \/>\n     &amp;= c + 0           &amp;&amp;\\text{[por suma con opuesto]} &#92;&#92;<br \/>\n     &amp;= c               &amp;&amp;\\text{[por suma con cero]}<br \/>\n\\end{align}<\/p>\n<p><b>2\u00aa demostraci\u00f3n en LN<\/b><\/p>\n<p>Por la siguiente cadena de igualdades<br \/>\n\\begin{align}<br \/>\n   a &amp;= (a + b) + -b    &#92;&#92;<br \/>\n     &amp;= (c + b) + -b    &amp;&amp;\\text{[por hip\u00f3tesis]} &#92;&#92;<br \/>\n     &amp;= c<br \/>\n\\end{align}<\/p>\n<p><b>Demostraciones con Lean4<\/b><\/p>\n<pre lang=\"lean\">\nimport Mathlib.Algebra.Ring.Defs\nimport Mathlib.Tactic\n\nvariable {R : Type _} [Ring R]\nvariable {a b c : R}\n\n-- 1\u00aa demostraci\u00f3n con Lean4\n-- =========================\n\nexample\n  (h : a + b = c + b)\n  : a = c :=\ncalc\n  a = a + 0        := by rw [add_zero]\n  _ = a + (b + -b) := by rw [add_right_neg]\n  _ = (a + b) + -b := by rw [add_assoc]\n  _ = (c + b) + -b := by rw [h]\n  _ = c + (b + -b) := by rw [\u2190 add_assoc]\n  _ = c + 0        := by rw [\u2190 add_right_neg]\n  _ = c            := by rw [add_zero]\n\n-- 2\u00aa demostraci\u00f3n con Lean4\n-- =========================\n\nexample\n  (h : a + b = c + b)\n  : a = c :=\ncalc\n  a = (a + b) + -b := (add_neg_cancel_right a b).symm\n  _ = (c + b) + -b := by rw [h]\n  _ = c            := add_neg_cancel_right c b\n\n-- 3\u00aa demostraci\u00f3n con Lean4\n-- =========================\n\nexample\n  (h : a + b = c + b)\n  : a = c :=\nby\n  rw [\u2190 add_neg_cancel_right a b]\n  rw [h]\n  rw [add_neg_cancel_right]\n\n-- 4\u00aa demostraci\u00f3n con Lean4\n-- =========================\n\nexample\n  (h : a + b = c + b)\n  : a = c :=\nby\n  rw [\u2190 add_neg_cancel_right a b, h, add_neg_cancel_right]\n\n-- 5\u00aa demostraci\u00f3n con Lean4\n-- =========================\n\nexample\n  (h : a + b = c + b)\n  : a = c :=\nadd_right_cancel h\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\/Cancelativa_derecha.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. 11.<\/li>\n<\/ul>\n<p><a name=\"ej4\"><\/a><\/p>\n<h3>4. Si R es un anillo y a \u2208 R, entonces a.0 = 0<\/h3>\n<p><em>Demostrar con Lean4 que si &#92;(R&#92;) es un anillo y &#92;(a &#92;in R&#92;), entonces &#92;(a\u00b70 = 0&#92;).<\/em><\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Algebra.Ring.Defs\nimport Mathlib.Tactic\n\nvariable {R : Type _} [Ring R]\nvariable (a : R)\n\nexample : a * 0 = 0 :=\nsorry\n<\/pre>\n<p><b>Demostraci\u00f3n en lenguaje natural<\/b><\/p>\n<p>Basta aplicar la propiedad cancelativa a<br \/>\n&#091;a.0 + a.0 = a.0 + 0&#093;<br \/>\nque se demuestra mediante la siguiente cadena de igualdades<br \/>\n\\begin{align}<br \/>\n   a.0 + a.0 &amp;= a.(0 + 0)    &amp;&amp;\\text{[por la distributiva]} &#92;&#92;<br \/>\n             &amp;= a.0          &amp;&amp;\\text{[por suma con cero]} &#92;&#92;<br \/>\n             &amp;= a.0 + 0      &amp;&amp;\\text{[por suma con cero]}<br \/>\n\\end{align}<\/p>\n<p><b>Demostraciones con Lean4<\/b><\/p>\n<pre lang=\"lean\">\nimport Mathlib.Algebra.Ring.Defs\nimport Mathlib.Tactic\n\nvariable {R : Type _} [Ring R]\nvariable (a : R)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : a * 0 = 0 :=\nby\n  have h : a * 0 + a * 0 = a * 0 + 0 :=\n    calc a * 0 + a * 0 = a * (0 + 0) := by rw [mul_add a 0 0]\n                     _ = a * 0       := by rw [add_zero 0]\n                     _ = a * 0 + 0   := by rw [add_zero (a * 0)]\n  rw [add_left_cancel h]\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : a * 0 = 0 :=\nby\n  have h : a * 0 + a * 0 = a * 0 + 0 :=\n    calc a * 0 + a * 0 = a * (0 + 0) := by rw [\u2190 mul_add]\n                     _ = a * 0       := by rw [add_zero]\n                     _ = a * 0 + 0   := by rw [add_zero]\n  rw [add_left_cancel h]\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : a * 0 = 0 :=\nby\n  have h : a * 0 + a * 0 = a * 0 + 0 :=\n    by rw [\u2190 mul_add, add_zero, add_zero]\n  rw [add_left_cancel h]\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : a * 0 = 0 :=\nby\n  have : a * 0 + a * 0 = a * 0 + 0 :=\n    calc a * 0 + a * 0 = a * (0 + 0) := by simp\n                     _ = a * 0       := by simp\n                     _ = a * 0 + 0   := by simp\n  simp\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample : a * 0 = 0 :=\n  mul_zero a\n\n-- 6\u00aa demostraci\u00f3n\n-- ===============\n\nexample : a * 0 = 0 :=\nby simp\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\/Multiplicacion_por_cero.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. 11.<\/li>\n<\/ul>\n<p><a name=\"ej5\"><\/a><\/p>\n<h3>5. Si R es un anillo y a \u2208 R, entonces 0.a = 0<\/h3>\n<p><em>Demostrar con Lean4 que si &#92;(R&#92;) es un anillo y &#92;(a &#92;in R&#92;), entonces &#92;(0\u00b7a = 0&#92;).<\/em><\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Algebra.Ring.Defs\nimport Mathlib.Tactic\n\nvariable {R : Type _} [Ring R]\nvariable (a : R)\n\nexample : 0 * a = 0 :=\nsorry\n<\/pre>\n<p><b>Demostraci\u00f3n en lenguaje natural<\/b><\/p>\n<p>Basta aplicar la propiedad cancelativa a<br \/>\n&#92;[0.a + 0.a = 0.a + 0&#92;]<br \/>\nque se demuestra mediante la siguiente cadena de igualdades<br \/>\n\\begin{align}<br \/>\n   0.a + 0.a &amp;= (0 + 0).a    &amp;&amp;\\text{[por la distributiva]} &#92;&#92;<br \/>\n             &amp;= 0.a          &amp;&amp;\\text{[por suma con cero]} &#92;&#92;<br \/>\n             &amp;= 0.a + 0      &amp;&amp;\\text{[por suma con cero]}<br \/>\n\\end{align}<\/p>\n<p><b>Demostraciones con Lean4<\/b><\/p>\n<pre lang=\"lean\">\nimport Mathlib.Algebra.Ring.Defs\nimport Mathlib.Tactic\n\nvariable {R : Type _} [Ring R]\nvariable (a : R)\n\n-- 1\u00aa demostraci\u00f3n\nexample : 0 * a = 0 :=\nby\n  have h : 0 * a + 0 * a = 0 * a + 0 :=\n    calc 0 * a + 0 * a = (0 + 0) * a := by rw [add_mul]\n                     _ = 0 * a       := by rw [add_zero]\n                     _ = 0 * a + 0   := by rw [add_zero]\n  rw [add_left_cancel h]\n\n-- 2\u00aa demostraci\u00f3n\nexample : 0 * a = 0 :=\nby\n  have h : 0 * a + 0 * a = 0 * a + 0 :=\n    by rw [\u2190add_mul, add_zero, add_zero]\n  rw [add_left_cancel h]\n\n-- 3\u00aa demostraci\u00f3n\nexample : 0 * a = 0 :=\nby\n  have : 0 * a + 0 * a = 0 * a + 0 :=\n    calc 0 * a + 0 * a = (0 + 0) * a := by simp\n                     _ = 0 * a       := by simp\n                     _ = 0 * a + 0   := by simp\n  simp\n\n-- 4\u00aa demostraci\u00f3n\nexample : 0 * a = 0 :=\nby\n  have : 0 * a + 0 * a = 0 * a + 0 := by simp\n  simp\n\n-- 5\u00aa demostraci\u00f3n\nexample : 0 * a = 0 :=\nby simp\n\n-- 6\u00aa demostraci\u00f3n\nexample : 0 * a = 0 :=\nzero_mul 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\/Multiplicacion_por_cero_izquierda.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. 11.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Esta semana he publicado en Calculemus las demostraciones con Lean4 de las siguientes propiedades: 1. Si R es un anillo y a, b \u2208 R, entonces (a + b) + -b = a 2. Si R es un anillo y a, b, c \u2208 R tales que a+b=a+c, entonces b=c 3. Si R es un&#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\/7985"}],"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=7985"}],"version-history":[{"count":15,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7985\/revisions"}],"predecessor-version":[{"id":8000,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7985\/revisions\/8000"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/media?parent=7985"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/categories?post=7985"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/tags?post=7985"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}