{"id":8152,"date":"2024-03-02T12:15:05","date_gmt":"2024-03-02T11:15:05","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/?p=8152"},"modified":"2024-03-02T12:15:05","modified_gmt":"2024-03-02T11:15:05","slug":"02-mar-24","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/02-mar-24\/","title":{"rendered":"La semana en Calculemus (2 de marzo de 2024)"},"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. s \\ (t \u222a u) \u2286 (s \\ t) \\ u<\/a><\/li>\n<li><a href=\"#ej2\">2. s \u2229 t = t \u2229 s<\/a><\/li>\n<li><a href=\"#ej3\">3. s \u2229 (s \u222a t) = s<\/a><\/li>\n<li><a href=\"#ej4\">4. s \u222a (s \u2229 t) = s<\/a><\/li>\n<li><a href=\"#ej5\">5. (s \\ t) \u222a t = s \u222a t<\/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. s \\ (t \u222a u) \u2286 (s \\ t) \\ u<\/h3>\n<p>Demostrar con Lean4 que<br \/>\n&#92;[ s \\setminus (t \u222a u) \u2286 (s \\setminus t) \\setminus u &#92;]<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Basic\nopen Set\nvariable {\u03b1 : Type}\nvariable (s t u : Set \u03b1)\n\nexample : s \\ (t \u222a u) \u2286 (s \\ t) \\ u :=\nby sorry\n<\/pre>\n<h2>1. Demostraci\u00f3n en lenguaje natural<\/h2>\n<p>Sea &#92;(x \u2208 s \\setminus (t \u222a u)&#92;). Entonces,<br \/>\n&#92;begin{align}<br \/>\n   &amp;x \u2208 s      &#92;tag{1} &#92;&#92;<br \/>\n   &amp;x \u2209 t \u222a u  &#92;tag{2} &#92;&#92;<br \/>\n&#92;end{align}<br \/>\nTenemos que demostrar que &#92;(x \u2208 (s \\setminus t) \\setminus u&#92;); es decir, que se verifican las relaciones<br \/>\n&#92;begin{align}<br \/>\n   &amp;x \u2208 s \\setminus t &#92;tag{3} &#92;&#92;<br \/>\n   &amp;x \u2209 u     &#92;tag{4}<br \/>\n&#92;end{align}<br \/>\nPara demostrar (3) tenemos que demostrar las relaciones<br \/>\n&#92;begin{align}<br \/>\n   &amp;x \u2208 s &#92;tag{5} &#92;&#92;<br \/>\n   &amp;x \u2209 t &#92;tag{6}<br \/>\n&#92;end{align}<br \/>\nLa (5) se tiene por la (1). Para demostrar la (6), supongamos que &#92;(x \u2208 t&#92;); entonces, &#92;(x \u2208 t \u222a u&#92;), en contracci\u00f3n con (2). Para demostrar la (4), supongamos que &#92;(x \u2208 u&#92;); entonces, &#92;(x \u2208 t \u222a u&#92;), en contracci\u00f3n con (2).<\/p>\n<h2>2. Demostraciones con Lean4<\/h2>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Basic\nopen Set\n\nvariable {\u03b1 : Type}\nvariable (s t u : Set \u03b1)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \\ (t \u222a u) \u2286 (s \\ t) \\ u :=\nby\n  intros x hx\n  -- x : \u03b1\n  -- hx : x \u2208 s \\ (t \u222a u)\n  -- \u22a2 x \u2208 (s \\ t) \\ u\n  constructor\n  . -- \u22a2 x \u2208 s \\ t\n    constructor\n    . -- \u22a2 x \u2208 s\n      exact hx.1\n    . -- \u22a2 \u00acx \u2208 t\n      intro xt\n      -- xt : x \u2208 t\n      -- \u22a2 False\n      apply hx.2\n      -- \u22a2 x \u2208 t \u222a u\n      left\n      -- \u22a2 x \u2208 t\n      exact xt\n  . -- \u22a2 \u00acx \u2208 u\n    intro xu\n    -- xu : x \u2208 u\n    -- \u22a2 False\n    apply hx.2\n    -- \u22a2 x \u2208 t \u222a u\n    right\n    -- \u22a2 x \u2208 u\n    exact xu\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \\ (t \u222a u) \u2286 (s \\ t) \\ u :=\nby\n  rintro x \u27e8xs, xntu\u27e9\n  -- x : \u03b1\n  -- xs : x \u2208 s\n  -- xntu : \u00acx \u2208 t \u222a u\n  -- \u22a2 x \u2208 (s \\ t) \\ u\n  constructor\n  . -- \u22a2 x \u2208 s \\ t\n    constructor\n    . -- \u22a2 x \u2208 s\n      exact xs\n    . -- \u00acx \u2208 t\n      intro xt\n      -- xt : x \u2208 t\n      -- \u22a2 False\n      exact xntu (Or.inl xt)\n  . -- \u22a2 \u00acx \u2208 u\n    intro xu\n    -- xu : x \u2208 u\n    -- \u22a2 False\n    exact xntu (Or.inr xu)\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \\ (t \u222a u) \u2286 (s \\ t) \\ u :=\n  fun _ \u27e8xs, xntu\u27e9 \u21a6 \u27e8\u27e8xs, fun xt \u21a6 xntu (Or.inl xt)\u27e9,\n                      fun xu \u21a6 xntu (Or.inr xu)\u27e9\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \\ (t \u222a u) \u2286 (s \\ t) \\ u :=\nby\n  rintro x \u27e8xs, xntu\u27e9\n  -- x : \u03b1\n  -- xs : x \u2208 s\n  -- xntu : \u00acx \u2208 t \u222a u\n  -- \u22a2 x \u2208 (s \\ t) \\ u\n  aesop\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \\ (t \u222a u) \u2286 (s \\ t) \\ u :=\nby intro ; aesop\n\n-- 6\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \\ (t \u222a u) \u2286 (s \\ t) \\ u :=\nby rw [diff_diff]\n\n-- Lema usado\n-- ==========\n\n-- #check (diff_diff : (s \\ t) \\ u = s \\ (t \u222a u))\n<\/pre>\n<p>Se puede interactuar con las demostraciones anteriores en <a href=\"https:\/\/live.lean-lang.org\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus2\/main\/src\/Diferencia_de_diferencia_de_conjuntos_2.lean\" rel=\"noopener noreferrer\" target=\"_blank\">Lean 4 Web<\/a>.<\/p>\n<h2>3. Demostraciones con Isabelle\/HOL<\/h2>\n<pre lang=\"isar\">\ntheory Diferencia_de_diferencia_de_conjuntos_2\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\nend\n<\/pre>\n<p><a name=\"ej2\"><\/a><\/p>\n<h3>2. s \u2229 t = t \u2229 s<\/h3>\n<p>Demostrar con Lean4 que<br \/>\n&#92;[ s \u2229 t = t \u2229 s &#92;]<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Basic\nopen Set\nvariable {\u03b1 : Type}\nvariable (s t : Set \u03b1)\n\nexample : s \u2229 t = t \u2229 s :=\nby sorry\n<\/pre>\n<h2>1. Demostraci\u00f3n en lenguaje natural<\/h2>\n<p>Tenemos que demostrar que<br \/>\n&#92;[ (\u2200 x)[x \u2208 s \u2229 t \u2194 x \u2208 t \u2229 s] &#92;]<br \/>\nDemostratemos la equivalencia por la doble implicaci\u00f3n.<\/p>\n<p>Sea &#92;(x \u2208 s \u2229 t&#92;). Entonces, se tiene<br \/>\n&#92;begin{align}<br \/>\n   &amp;x \u2208 s &#92;tag{1} &#92;&#92;<br \/>\n   &amp;x \u2208 t &#92;tag{2}<br \/>\n&#92;end{align}<br \/>\nLuego &#92;(x \u2208 t \u2229 s&#92;) (por (2) y (1)).<\/p>\n<p>La segunda implicaci\u00f3n se demuestra an\u00e1logamente.<\/p>\n<h2>2. Demostraciones con Lean4<\/h2>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Basic\nopen Set\n\nvariable {\u03b1 : Type}\nvariable (s t : Set \u03b1)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 t = t \u2229 s :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 s \u2229 t \u2194 x \u2208 t \u2229 s\n  simp only [mem_inter_iff]\n  -- \u22a2 x \u2208 s \u2227 x \u2208 t \u2194 x \u2208 t \u2227 x \u2208 s\n  constructor\n  . -- \u22a2 x \u2208 s \u2227 x \u2208 t \u2192 x \u2208 t \u2227 x \u2208 s\n    intro h\n    -- h : x \u2208 s \u2227 x \u2208 t\n    -- \u22a2 x \u2208 t \u2227 x \u2208 s\n    constructor\n    . -- \u22a2 x \u2208 t\n      exact h.2\n    . -- \u22a2 x \u2208 s\n      exact h.1\n  . -- \u22a2 x \u2208 t \u2227 x \u2208 s \u2192 x \u2208 s \u2227 x \u2208 t\n    intro h\n    -- h : x \u2208 t \u2227 x \u2208 s\n    -- \u22a2 x \u2208 s \u2227 x \u2208 t\n    constructor\n    . -- \u22a2 x \u2208 s\n      exact h.2\n    . -- \u22a2 x \u2208 t\n      exact h.1\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 t = t \u2229 s :=\nby\n  ext\n  -- x : \u03b1\n  -- \u22a2 x \u2208 s \u2229 t \u2194 x \u2208 t \u2229 s\n  simp only [mem_inter_iff]\n  -- \u22a2 x \u2208 s \u2227 x \u2208 t \u2194 x \u2208 t \u2227 x \u2208 s\n  exact \u27e8fun h \u21a6 \u27e8h.2, h.1\u27e9,\n         fun h \u21a6 \u27e8h.2, h.1\u27e9\u27e9\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 t = t \u2229 s :=\nby\n  ext\n  -- x : \u03b1\n  -- \u22a2 x \u2208 s \u2229 t \u2194 x \u2208 t \u2229 s\n  exact \u27e8fun h \u21a6 \u27e8h.2, h.1\u27e9,\n         fun h \u21a6 \u27e8h.2, h.1\u27e9\u27e9\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 t = t \u2229 s :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 s \u2229 t \u2194 x \u2208 t \u2229 s\n  simp only [mem_inter_iff]\n  -- \u22a2 x \u2208 s \u2227 x \u2208 t \u2194 x \u2208 t \u2227 x \u2208 s\n  constructor\n  . -- \u22a2 x \u2208 s \u2227 x \u2208 t \u2192 x \u2208 t \u2227 x \u2208 s\n    rintro \u27e8xs, xt\u27e9\n    -- xs : x \u2208 s\n    -- xt : x \u2208 t\n    -- \u22a2 x \u2208 t \u2227 x \u2208 s\n    exact \u27e8xt, xs\u27e9\n  . -- \u22a2 x \u2208 t \u2227 x \u2208 s \u2192 x \u2208 s \u2227 x \u2208 t\n    rintro \u27e8xt, xs\u27e9\n    -- xt : x \u2208 t\n    -- xs : x \u2208 s\n    -- \u22a2 x \u2208 s \u2227 x \u2208 t\n    exact \u27e8xs, xt\u27e9\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 t = t \u2229 s :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 s \u2229 t \u2194 x \u2208 t \u2229 s\n  simp only [mem_inter_iff]\n  -- \u22a2 x \u2208 s \u2227 x \u2208 t \u2194 x \u2208 t \u2227 x \u2208 s\n  simp only [And.comm]\n\n-- 6\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 t = t \u2229 s :=\next (fun _ \u21a6 And.comm)\n\n-- 7\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 t = t \u2229 s :=\nby ext ; simp [And.comm]\n\n-- 8\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 t = t \u2229 s :=\ninter_comm s t\n\n-- Lemas usados\n-- ============\n\n-- variable (x : \u03b1)\n-- variable (a b : Prop)\n-- #check (And.comm : a \u2227 b \u2194 b \u2227 a)\n-- #check (inter_comm s t : s \u2229 t = t \u2229 s)\n-- #check (mem_inter_iff x s t : x \u2208 s \u2229 t \u2194 x \u2208 s \u2227 x \u2208 t)\n<\/pre>\n<p>Se puede interactuar con las demostraciones anteriores en <a href=\"https:\/\/live.lean-lang.org\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus2\/main\/src\/Conmutatividad_de_la_interseccion.lean\" rel=\"noopener noreferrer\" target=\"_blank\">Lean 4 Web<\/a>.<\/p>\n<h2>3. Demostraciones con Isabelle\/HOL<\/h2>\n<pre lang=\"isar\">\ntheory Conmutatividad_de_la_interseccion\nimports Main\nbegin\n\n(* 1\u00aa demostraci\u00f3n *)\nlemma \"s \u2229 t = t \u2229 s\"\nproof (rule set_eqI)\n  fix x\n  show \"x \u2208 s \u2229 t \u27f7 x \u2208 t \u2229 s\"\n  proof (rule iffI)\n    assume h : \"x \u2208 s \u2229 t\"\n    then have xs : \"x \u2208 s\"\n      by (simp only: IntD1)\n    have xt : \"x \u2208 t\"\n      using h by (simp only: IntD2)\n    then show \"x \u2208 t \u2229 s\"\n      using xs by (rule IntI)\n  next\n    assume h : \"x \u2208 t \u2229 s\"\n    then have xt : \"x \u2208 t\"\n      by (simp only: IntD1)\n    have xs : \"x \u2208 s\"\n      using h by (simp only: IntD2)\n    then show \"x \u2208 s \u2229 t\"\n      using xt by (rule IntI)\n  qed\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\nlemma \"s \u2229 t = t \u2229 s\"\nproof (rule set_eqI)\n  fix x\n  show \"x \u2208 s \u2229 t \u27f7 x \u2208 t \u2229 s\"\n  proof\n    assume h : \"x \u2208 s \u2229 t\"\n    then have xs : \"x \u2208 s\"\n      by simp\n    have xt : \"x \u2208 t\"\n      using h by simp\n    then show \"x \u2208 t \u2229 s\"\n      using xs by simp\n  next\n    assume h : \"x \u2208 t \u2229 s\"\n    then have xt : \"x \u2208 t\"\n      by simp\n    have xs : \"x \u2208 s\"\n      using h by simp\n    then show \"x \u2208 s \u2229 t\"\n      using xt by simp\n  qed\nqed\n\n(* 3\u00aa demostraci\u00f3n *)\nlemma \"s \u2229 t = t \u2229 s\"\nproof (rule equalityI)\n  show \"s \u2229 t \u2286 t \u2229 s\"\n  proof (rule subsetI)\n    fix x\n    assume h : \"x \u2208 s \u2229 t\"\n    then have xs : \"x \u2208 s\"\n      by (simp only: IntD1)\n    have xt : \"x \u2208 t\"\n      using h by (simp only: IntD2)\n    then show \"x \u2208 t \u2229 s\"\n      using xs by (rule IntI)\n  qed\nnext\n  show \"t \u2229 s \u2286 s \u2229 t\"\n  proof (rule subsetI)\n    fix x\n    assume h : \"x \u2208 t \u2229 s\"\n    then have xt : \"x \u2208 t\"\n      by (simp only: IntD1)\n    have xs : \"x \u2208 s\"\n      using h by (simp only: IntD2)\n    then show \"x \u2208 s \u2229 t\"\n      using xt by (rule IntI)\n  qed\nqed\n\n(* 4\u00aa demostraci\u00f3n *)\nlemma \"s \u2229 t = t \u2229 s\"\nproof\n  show \"s \u2229 t \u2286 t \u2229 s\"\n  proof\n    fix x\n    assume h : \"x \u2208 s \u2229 t\"\n    then have xs : \"x \u2208 s\"\n      by simp\n    have xt : \"x \u2208 t\"\n      using h by simp\n    then show \"x \u2208 t \u2229 s\"\n      using xs by simp\n  qed\nnext\n  show \"t \u2229 s \u2286 s \u2229 t\"\n  proof\n    fix x\n    assume h : \"x \u2208 t \u2229 s\"\n    then have xt : \"x \u2208 t\"\n      by simp\n    have xs : \"x \u2208 s\"\n      using h by simp\n    then show \"x \u2208 s \u2229 t\"\n      using xt by simp\n  qed\nqed\n\n(* 5\u00aa demostraci\u00f3n *)\nlemma \"s \u2229 t = t \u2229 s\"\nproof\n  show \"s \u2229 t \u2286 t \u2229 s\"\n  proof\n    fix x\n    assume \"x \u2208 s \u2229 t\"\n    then show \"x \u2208 t \u2229 s\"\n      by simp\n  qed\nnext\n  show \"t \u2229 s \u2286 s \u2229 t\"\n  proof\n    fix x\n    assume \"x \u2208 t \u2229 s\"\n    then show \"x \u2208 s \u2229 t\"\n      by simp\n  qed\nqed\n\n(* 6\u00aa demostraci\u00f3n *)\nlemma \"s \u2229 t = t \u2229 s\"\nby (fact Int_commute)\n\n(* 7\u00aa demostraci\u00f3n *)\nlemma \"s \u2229 t = t \u2229 s\"\nby (fact inf_commute)\n\n(* 8\u00aa demostraci\u00f3n *)\nlemma \"s \u2229 t = t \u2229 s\"\nby auto\n\nend\n<\/pre>\n<p><a name=\"ej3\"><\/a><\/p>\n<h3>3. s \u2229 (s \u222a t) = s<\/h3>\n<p>Demostrar con Lean4 que<br \/>\n&#92;[ s \u2229 (s \u222a t) = s &#92;]<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Basic\nimport Mathlib.Tactic\nopen Set\nvariable {\u03b1 : Type}\nvariable (s t : Set \u03b1)\n\nexample : s \u2229 (s \u222a t) = s :=\nby sorry\n<\/pre>\n<h2>1. Demostraci\u00f3n en lenguaje natural<\/h2>\n<p>Tenemos que demostrar que<br \/>\n&#92;[ (\u2200 x)[x \u2208 s \u2229 (s \u222a t) \u2194 x \u2208 s] &#92;]<br \/>\ny lo haremos demostrando las dos implicaciones.<\/p>\n<p>(\u27f9) Sea &#92;(x \u2208 s \u2229 (s \u222a t)&#92;). Entonces, &#92;(x \u2208 s&#92;).<\/p>\n<p>(\u27f8) Sea &#92;(x \u2208 s&#92;). Entonces, &#92;(x \u2208 s \u222a t&#92;) y, por tanto, &#92;(x \u2208 s \u2229 (s \u222a t)&#92;).<\/p>\n<h2>2. Demostraciones con Lean4<\/h2>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Basic\nimport Mathlib.Tactic\nopen Set\n\nvariable {\u03b1 : Type}\nvariable (s t : Set \u03b1)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (s \u222a t) = s :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 s \u2229 (s \u222a t) \u2194 x \u2208 s\n  constructor\n  . -- \u22a2 x \u2208 s \u2229 (s \u222a t) \u2192 x \u2208 s\n    intros h\n  -- h : x \u2208 s \u2229 (s \u222a t)\n  -- \u22a2 x \u2208 s\n    exact h.1\n  . -- \u22a2 x \u2208 s \u2192 x \u2208 s \u2229 (s \u222a t)\n    intro xs\n    -- xs : x \u2208 s\n    -- \u22a2 x \u2208 s \u2229 (s \u222a t)\n    constructor\n    . -- \u22a2 x \u2208 s\n      exact xs\n    . -- \u22a2 x \u2208 s \u222a t\n      left\n      -- \u22a2 x \u2208 s\n      exact xs\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (s \u222a t) = s :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 s \u2229 (s \u222a t) \u2194 x \u2208 s\n  constructor\n  . -- \u22a2 x \u2208 s \u2229 (s \u222a t) \u2192 x \u2208 s\n    intro h\n    -- h : x \u2208 s \u2229 (s \u222a t)\n    -- \u22a2 x \u2208 s\n    exact h.1\n  . -- \u22a2 x \u2208 s \u2192 x \u2208 s \u2229 (s \u222a t)\n    intro xs\n    -- xs : x \u2208 s\n    -- \u22a2 x \u2208 s \u2229 (s \u222a t)\n    constructor\n    . -- \u22a2 x \u2208 s\n      exact xs\n    . -- \u22a2 x \u2208 s \u222a t\n      exact (Or.inl xs)\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (s \u222a t) = s :=\nby\n  ext\n  -- x : \u03b1\n  -- \u22a2 x \u2208 s \u2229 (s \u222a t) \u2194 x \u2208 s\n  exact \u27e8fun h \u21a6 h.1,\n         fun xs \u21a6 \u27e8xs, Or.inl xs\u27e9\u27e9\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (s \u222a t) = s :=\nby\n  ext\n  -- x : \u03b1\n  -- \u22a2 x \u2208 s \u2229 (s \u222a t) \u2194 x \u2208 s\n  exact \u27e8And.left,\n         fun xs \u21a6 \u27e8xs, Or.inl xs\u27e9\u27e9\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (s \u222a t) = s :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 s \u2229 (s \u222a t) \u2194 x \u2208 s\n  constructor\n  . -- \u22a2 x \u2208 s \u2229 (s \u222a t) \u2192 x \u2208 s\n    rintro \u27e8xs, -\u27e9\n    -- xs : x \u2208 s\n    -- \u22a2 x \u2208 s\n    exact xs\n  . -- \u22a2 x \u2208 s \u2192 x \u2208 s \u2229 (s \u222a t)\n    intro xs\n    -- xs : x \u2208 s\n    -- \u22a2 x \u2208 s \u2229 (s \u222a t)\n    use xs\n    -- \u22a2 x \u2208 s \u222a t\n    left\n    -- \u22a2 x \u2208 s\n    exact xs\n\n-- 6\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (s \u222a t) = s :=\nby\n  apply subset_antisymm\n  . -- \u22a2 s \u2229 (s \u222a t) \u2286 s\n    rintro x \u27e8hxs, -\u27e9\n    -- x : \u03b1\n    -- hxs : x \u2208 s\n    -- \u22a2 x \u2208 s\n    exact hxs\n  . -- \u22a2 s \u2286 s \u2229 (s \u222a t)\n    intros x hxs\n    -- x : \u03b1\n    -- hxs : x \u2208 s\n    -- \u22a2 x \u2208 s \u2229 (s \u222a t)\n    exact \u27e8hxs, Or.inl hxs\u27e9\n\n-- 7\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (s \u222a t) = s :=\ninf_sup_self\n\n-- 8\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (s \u222a t) = s :=\nby aesop\n\n-- Lemas usados\n-- ============\n\n-- variable (a b : Prop)\n-- #check (And.left : a \u2227 b \u2192 a)\n-- #check (Or.inl : a \u2192 a \u2228 b)\n-- #check (inf_sup_self : s \u2229 (s \u222a t) = s)\n-- #check (subset_antisymm : s \u2286 t \u2192 t \u2286 s \u2192 s = t)\n<\/pre>\n<p>Se puede interactuar con las demostraciones anteriores en <a href=\"https:\/\/live.lean-lang.org\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus2\/main\/src\/Interseccion_con_su_union.lean\" rel=\"noopener noreferrer\" target=\"_blank\">Lean 4 Web<\/a>.<\/p>\n<h2>3. Demostraciones con Isabelle\/HOL<\/h2>\n<pre lang=\"isar\">\ntheory Interseccion_con_su_union\nimports Main\nbegin\n\n(* 1\u00aa demostraci\u00f3n *)\nlemma \"s \u2229 (s \u222a t) = s\"\nproof (rule  equalityI)\n  show \"s \u2229 (s \u222a t) \u2286 s\"\n  proof (rule subsetI)\n    fix x\n    assume \"x \u2208 s \u2229 (s \u222a t)\"\n    then show \"x \u2208 s\"\n      by (simp only: IntD1)\n  qed\nnext\n  show \"s \u2286 s \u2229 (s \u222a t)\"\n  proof (rule subsetI)\n    fix x\n    assume \"x \u2208 s\"\n    then have \"x \u2208 s \u222a t\"\n      by (simp only: UnI1)\n    with \u2039x \u2208 s\u203a show \"x \u2208 s \u2229 (s \u222a t)\"\n      by (rule IntI)\n  qed\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\nlemma \"s \u2229 (s \u222a t) = s\"\nproof\n  show \"s \u2229 (s \u222a t) \u2286 s\"\n  proof\n    fix x\n    assume \"x \u2208 s \u2229 (s \u222a t)\"\n    then show \"x \u2208 s\"\n      by simp\n  qed\nnext\n  show \"s \u2286 s \u2229 (s \u222a t)\"\n  proof\n    fix x\n    assume \"x \u2208 s\"\n    then have \"x \u2208 s \u222a t\"\n      by simp\n    then show \"x \u2208 s \u2229 (s \u222a t)\"\n      using \u2039x \u2208 s\u203a by simp\n  qed\nqed\n\n(* 3\u00aa demostraci\u00f3n *)\nlemma \"s \u2229 (s \u222a t) = s\"\nby (fact Un_Int_eq)\n\n(* 4\u00aa demostraci\u00f3n *)\nlemma \"s \u2229 (s \u222a t) = s\"\nby auto\n<\/pre>\n<p><a name=\"ej4\"><\/a><\/p>\n<h3>4. s \u222a (s \u2229 t) = s<\/h3>\n<p>Demostrar con Lean4 que<br \/>\n&#92;[ s \u222a (s \u2229 t) = s &#92;]<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Basic\nopen Set\nvariable {\u03b1 : Type}\nvariable (s t : Set \u03b1)\n\nexample : s \u222a (s \u2229 t) = s :=\nby sorry\n<\/pre>\n<h2>1. Demostraci\u00f3n en lenguaje natural<\/h2>\n<p>Tenemos que demostrar que<br \/>\n&#92;[ (\u2200 x)[x \u2208 s \u222a (s \u2229 t) \u2194 x \u2208 s] &#92;]<br \/>\ny lo haremos demostrando las dos implicaciones.<\/p>\n<p>(\u27f9) Sea &#92;(x \u2208 s \u222a (s \u2229 t)&#92;). Entonces, &#92;(x \u2208 s&#92;) \u00f3 &#92;(x \u2208 s \u2229 t&#92;). En ambos casos, &#92;(x \u2208 s&#92;).<\/p>\n<p>(\u27f8) Sea &#92;(x \u2208 s&#92;). Entonces, &#92;(x \u2208 s \u2229 t&#92;) y, por tanto, &#92;(x \u2208 s \u222a (s \u2229 t)&#92;).<\/p>\n<h2>2. Demostraciones con Lean4<\/h2>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Basic\nopen Set\n\nvariable {\u03b1 : Type}\nvariable (s t : Set \u03b1)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u222a (s \u2229 t) = s :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 s \u222a (s \u2229 t) \u2194 x \u2208 s\n  constructor\n  . -- \u22a2 x \u2208 s \u222a (s \u2229 t) \u2192 x \u2208 s\n    intro hx\n    -- hx : x \u2208 s \u222a (s \u2229 t)\n    -- \u22a2 x \u2208 s\n    rcases hx with (xs | xst)\n    . -- xs : x \u2208 s\n      exact xs\n    . -- xst : x \u2208 s \u2229 t\n      exact xst.1\n  . -- \u22a2 x \u2208 s \u2192 x \u2208 s \u222a (s \u2229 t)\n    intro xs\n    -- xs : x \u2208 s\n    -- \u22a2 x \u2208 s \u222a (s \u2229 t)\n    left\n    -- \u22a2 x \u2208 s\n    exact xs\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u222a (s \u2229 t) = s :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 s \u222a s \u2229 t \u2194 x \u2208 s\n  exact \u27e8fun hx \u21a6 Or.elim hx id And.left,\n         fun xs \u21a6 Or.inl xs\u27e9\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u222a (s \u2229 t) = s :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 s \u222a (s \u2229 t) \u2194 x \u2208 s\n  constructor\n  . -- \u22a2 x \u2208 s \u222a (s \u2229 t) \u2192 x \u2208 s\n    rintro (xs | \u27e8xs, -\u27e9) <;>\n    -- xs : x \u2208 s\n    -- \u22a2 x \u2208 s\n    exact xs\n  . -- \u22a2 x \u2208 s \u2192 x \u2208 s \u222a (s \u2229 t)\n    intro xs\n    -- xs : x \u2208 s\n    -- \u22a2 x \u2208 s \u222a s \u2229 t\n    left\n    -- \u22a2 x \u2208 s\n    exact xs\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u222a (s \u2229 t) = s :=\nsup_inf_self\n\n-- Lemas usados\n-- ============\n\n-- variable (a b c : Prop)\n-- #check (And.left : a \u2227 b \u2192 a)\n-- #check (Or.elim : a \u2228 b \u2192 (a \u2192 c) \u2192 (b \u2192 c) \u2192 c)\n-- #check (sup_inf_self : s \u222a (s \u2229 t) = s)\n<\/pre>\n<p>Se puede interactuar con las demostraciones anteriores en <a href=\"https:\/\/live.lean-lang.org\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus2\/main\/src\/Union_con_su_interseccion.lean\" rel=\"noopener noreferrer\" target=\"_blank\">Lean 4 Web<\/a>.<\/p>\n<h2>3. Demostraciones con Isabelle\/HOL<\/h2>\n<pre lang=\"isar\">\ntheory Union_con_su_interseccion\nimports Main\nbegin\n\n(* 1\u00aa demostraci\u00f3n *)\nlemma \"s \u222a (s \u2229 t) = s\"\nproof (rule equalityI)\n  show \"s \u222a (s \u2229 t) \u2286 s\"\n  proof (rule subsetI)\n    fix x\n    assume \"x \u2208 s \u222a (s \u2229 t)\"\n    then show \"x \u2208 s\"\n    proof\n      assume \"x \u2208 s\"\n      then show \"x \u2208 s\"\n        by this\n    next\n      assume \"x \u2208 s \u2229 t\"\n      then show \"x \u2208 s\"\n        by (simp only: IntD1)\n    qed\n  qed\nnext\n  show \"s \u2286 s \u222a (s \u2229 t)\"\n  proof (rule subsetI)\n    fix x\n    assume \"x \u2208 s\"\n    then show \"x \u2208 s \u222a (s \u2229 t)\"\n      by (simp only: UnI1)\n  qed\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\nlemma \"s \u222a (s \u2229 t) = s\"\nproof\n  show \"s \u222a s \u2229 t \u2286 s\"\n  proof\n    fix x\n    assume \"x \u2208 s \u222a (s \u2229 t)\"\n    then show \"x \u2208 s\"\n    proof\n      assume \"x \u2208 s\"\n      then show \"x \u2208 s\"\n        by this\n    next\n      assume \"x \u2208 s \u2229 t\"\n      then show \"x \u2208 s\"\n        by simp\n    qed\n  qed\nnext\n  show \"s \u2286 s \u222a (s \u2229 t)\"\n  proof\n    fix x\n    assume \"x \u2208 s\"\n    then show \"x \u2208 s \u222a (s \u2229 t)\"\n      by simp\n  qed\nqed\n\n(* 3\u00aa demostraci\u00f3n *)\nlemma \"s \u222a (s \u2229 t) = s\"\n  by auto\n\nend\n<\/pre>\n<p><a name=\"ej5\"><\/a><\/p>\n<h3>5. (s \\ t) \u222a t = s \u222a t<\/h3>\n<p>Demostrar con Lean4 que<br \/>\n&#92;[ (s &#92;setminus t) \u222a t = s \u222a t &#92;]<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Basic\nopen Set\n\nvariable {\u03b1 : Type}\nvariable (s t : Set \u03b1)\n\nexample : (s \\\\setminus t) \u222a t = s \u222a t :=\nby sorry\n<\/pre>\n<h2>1. Demostraci\u00f3n en lenguaje natural<\/h2>\n<p>Tenemos que demostrar que<br \/>\n&#92;[ (\u2200 x)[x \u2208 (s &#92;setminus t) \u222a t \u2194 x \u2208 s \u222a t] &#92;]<br \/>\ny lo demostraremos por la siguiente cadena de equivalencias:<br \/>\n&#92;begin{align}<br \/>\n   x \u2208 (s &#92;setminus t) \u222a t<br \/>\n                   &amp;\u2194 x \u2208 (s &#92;setminus t) \u2228 (x \u2208 t)             &#92;&#92;<br \/>\n                   &amp;\u2194 (x \u2208 s \u2227 x \u2209 t) \u2228 x \u2208 t           &#92;&#92;<br \/>\n                   &amp;\u2194 (x \u2208 s \u2228 x \u2208 t) \u2227 (x \u2209 t \u2228 x \u2208 t) &#92;&#92;<br \/>\n                   &amp;\u2194 x \u2208 s \u2228 x \u2208 t                     &#92;&#92;<br \/>\n                   &amp;\u2194 x \u2208 s \u222a t<br \/>\n&#92;end{align}<\/p>\n<h2>2. Demostraciones con Lean4<\/h2>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Basic\nopen Set\n\nvariable {\u03b1 : Type}\nvariable (s t : Set \u03b1)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (s \\ t) \u222a t = s \u222a t :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 (s \\ t) \u222a t \u2194 x \u2208 s \u222a t\n  calc x \u2208 (s \\ t) \u222a t\n       \u2194 x \u2208 s \\ t \u2228 x \u2208 t                 := mem_union x (s \\ t) t\n     _ \u2194 (x \u2208 s \u2227 x \u2209 t) \u2228 x \u2208 t           := by simp only [mem_diff x]\n     _ \u2194 (x \u2208 s \u2228 x \u2208 t) \u2227 (x \u2209 t \u2228 x \u2208 t) := and_or_right\n     _ \u2194 (x \u2208 s \u2228 x \u2208 t) \u2227 True            := by simp only [em' (x \u2208 t)]\n     _ \u2194 x \u2208 s \u2228 x \u2208 t                     := and_true_iff (x \u2208 s \u2228 x \u2208 t)\n     _ \u2194 x \u2208 s \u222a t                         := (mem_union x s t).symm\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (s \\ t) \u222a t = s \u222a t :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 (s \\ t) \u222a t \u2194 x \u2208 s \u222a t\n  constructor\n  . -- \u22a2 x \u2208 (s \\ t) \u222a t \u2192 x \u2208 s \u222a t\n    intro hx\n    -- hx : x \u2208 (s \\ t) \u222a t\n    -- \u22a2 x \u2208 s \u222a t\n    rcases hx with (xst | xt)\n    . -- xst : x \u2208 s \\ t\n      -- \u22a2 x \u2208 s \u222a t\n      left\n      -- \u22a2 x \u2208 s\n      exact xst.1\n    . -- xt : x \u2208 t\n      -- \u22a2 x \u2208 s \u222a t\n      right\n      -- \u22a2 x \u2208 t\n      exact xt\n  . -- \u22a2 x \u2208 s \u222a t \u2192 x \u2208 (s \\ t) \u222a t\n    by_cases h : x \u2208 t\n    . -- h : x \u2208 t\n      intro _xst\n      -- _xst : x \u2208 s \u222a t\n      right\n      -- \u22a2 x \u2208 t\n      exact h\n    . -- \u22a2 x \u2208 s \u222a t \u2192 x \u2208 (s \\ t) \u222a t\n      intro hx\n      -- hx : x \u2208 s \u222a t\n      -- \u22a2 x \u2208 (s \\ t) \u222a t\n      rcases hx with (xs | xt)\n      . -- xs : x \u2208 s\n        left\n        -- \u22a2 x \u2208 s \\ t\n        constructor\n        . -- \u22a2 x \u2208 s\n          exact xs\n        . -- \u22a2 \u00acx \u2208 t\n          exact h\n      . -- xt : x \u2208 t\n        right\n        -- \u22a2 x \u2208 t\n        exact xt\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (s \\ t) \u222a t = s \u222a t :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 (s \\ t) \u222a t \u2194 x \u2208 s \u222a t\n  constructor\n  . -- \u22a2 x \u2208 (s \\ t) \u222a t \u2192 x \u2208 s \u222a t\n    rintro (\u27e8xs, -\u27e9 | xt)\n    . -- xs : x \u2208 s\n      -- \u22a2 x \u2208 s \u222a t\n      left\n      -- \u22a2 x \u2208 s\n      exact xs\n    . -- xt : x \u2208 t\n      -- \u22a2 x \u2208 s \u222a t\n      right\n      -- \u22a2 x \u2208 t\n      exact xt\n  . -- \u22a2 x \u2208 s \u222a t \u2192 x \u2208 (s \\ t) \u222a t\n    by_cases h : x \u2208 t\n    . -- h : x \u2208 t\n      intro _xst\n      -- _xst : x \u2208 s \u222a t\n      -- \u22a2 x \u2208 (s \\ t) \u222a t\n      right\n      -- \u22a2 x \u2208 t\n      exact h\n    . -- \u22a2 x \u2208 s \u222a t \u2192 x \u2208 (s \\ t) \u222a t\n      rintro (xs | xt)\n      . -- xs : x \u2208 s\n        -- \u22a2 x \u2208 (s \\ t) \u222a t\n        left\n        -- \u22a2 x \u2208 s \\ t\n        exact \u27e8xs, h\u27e9\n      . -- xt : x \u2208 t\n        -- \u22a2 x \u2208 (s \\ t) \u222a t\n        right\n        -- \u22a2 x \u2208 t\n        exact xt\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (s \\ t) \u222a t = s \u222a t :=\ndiff_union_self\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (s \\ t) \u222a t = s \u222a t :=\nby\n  ext\n  -- x : \u03b1\n  -- \u22a2 x \u2208 s \\ t \u222a t \u2194 x \u2208 s \u222a t\n  simp\n\n-- 6\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (s \\ t) \u222a t = s \u222a t :=\nby simp\n\n-- Lemas usados\n-- ============\n\n-- variable (a b c : Prop)\n-- variable (x : \u03b1)\n-- #check (and_or_right : (a \u2227 b) \u2228 c \u2194 (a \u2228 c) \u2227 (b \u2228 c))\n-- #check (and_true_iff a : a \u2227 True \u2194 a)\n-- #check (diff_union_self : (s \\ t) \u222a t = s \u222a t)\n-- #check (em' a : \u00aca \u2228 a)\n-- #check (mem_diff x : x \u2208 s \\ t \u2194 x \u2208 s \u2227 x \u2209 t)\n-- #check (mem_union x s t : x \u2208 s \u222a t \u2194 x \u2208 s \u2228 x \u2208 t)\n<\/pre>\n<p>Se puede interactuar con las demostraciones anteriores en <a href=\"https:\/\/live.lean-lang.org\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus2\/main\/src\/(s \\ t) \u222a t = s \u222a t.\" rel=\"noopener noreferrer\" target=\"_blank\">Lean 4 Web<\/a>.<\/p>\n<h2>3. Demostraciones con Isabelle\/HOL<\/h2>\n<pre lang=\"isar\">\ntheory Union_con_su_diferencia\nimports Main\nbegin\n\n(* 1\u00aa demostraci\u00f3n *)\nlemma \"(s - t) \u222a t = s \u222a t\"\nproof (rule equalityI)\n  show \"(s - t) \u222a t \u2286 s \u222a t\"\n  proof (rule subsetI)\n    fix x\n    assume \"x \u2208 (s - t) \u222a t\"\n    then show \"x \u2208 s \u222a t\"\n    proof (rule UnE)\n      assume \"x \u2208 s - t\"\n      then have \"x \u2208 s\"\n        by (simp only: DiffD1)\n      then show \"x \u2208 s \u222a t\"\n        by (simp only: UnI1)\n    next\n      assume \"x \u2208 t\"\n      then show \"x \u2208 s \u222a t\"\n        by (simp only: UnI2)\n    qed\n  qed\nnext\n  show \"s \u222a t \u2286 (s - t) \u222a t\"\n  proof (rule subsetI)\n    fix x\n    assume \"x \u2208 s \u222a t\"\n    then show \"x \u2208 (s - t) \u222a t\"\n    proof (rule UnE)\n      assume \"x \u2208 s\"\n      show \"x \u2208 (s - t) \u222a t\"\n      proof (cases \u2039x \u2208 t\u203a)\n        assume \"x \u2208 t\"\n        then show \"x \u2208 (s - t) \u222a t\"\n          by (simp only: UnI2)\n      next\n        assume \"x \u2209 t\"\n        with \u2039x \u2208 s\u203a have \"x \u2208 s - t\"\n          by (rule DiffI)\n        then show \"x \u2208 (s - t) \u222a t\"\n          by (simp only: UnI1)\n      qed\n    next\n      assume \"x \u2208 t\"\n      then show \"x \u2208 (s - t) \u222a t\"\n        by (simp only: UnI2)\n    qed\n  qed\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\nlemma \"(s - t) \u222a t = s \u222a t\"\nproof\n  show \"(s - t) \u222a t \u2286 s \u222a t\"\n  proof\n    fix x\n    assume \"x \u2208 (s - t) \u222a t\"\n    then show \"x \u2208 s \u222a t\"\n    proof\n      assume \"x \u2208 s - t\"\n      then have \"x \u2208 s\"\n        by simp\n      then show \"x \u2208 s \u222a t\"\n        by simp\n    next\n      assume \"x \u2208 t\"\n      then show \"x \u2208 s \u222a t\"\n        by simp\n    qed\n  qed\nnext\n  show \"s \u222a t \u2286 (s - t) \u222a t\"\n  proof\n    fix x\n    assume \"x \u2208 s \u222a t\"\n    then show \"x \u2208 (s - t) \u222a t\"\n    proof\n      assume \"x \u2208 s\"\n      show \"x \u2208 (s - t) \u222a t\"\n      proof\n        assume \"x \u2209 t\"\n        with \u2039x \u2208 s\u203a show \"x \u2208 s - t\"\n          by simp\n      qed\n    next\n      assume \"x \u2208 t\"\n      then show \"x \u2208 (s - t) \u222a t\"\n        by simp\n    qed\n  qed\nqed\n\n(* 3\u00aa demostraci\u00f3n *)\nlemma \"(s - t) \u222a t = s \u222a t\"\nby (fact Un_Diff_cancel2)\n\n(* 4\u00aa demostraci\u00f3n *)\nlemma \"(s - t) \u222a t = s \u222a t\"\n  by auto\n\nend\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Esta semana he publicado en Calculemus las demostraciones con Lean4 de las siguientes propiedades: 1. s \\ (t \u222a u) \u2286 (s \\ t) \\ u 2. s \u2229 t = t \u2229 s 3. s \u2229 (s \u222a t) = s 4. s \u222a (s \u2229 t) = s 5. (s \\ t) \u222a&#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":"default","_kad_post_title":"default","_kad_post_layout":"default","_kad_post_sidebar_id":"","_kad_post_content_style":"default","_kad_post_vertical_padding":"default","_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\/8152"}],"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=8152"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/8152\/revisions"}],"predecessor-version":[{"id":8153,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/8152\/revisions\/8153"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/media?parent=8152"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/categories?post=8152"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/tags?post=8152"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}