{"id":8155,"date":"2024-03-09T12:14:49","date_gmt":"2024-03-09T11:14:49","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/?p=8155"},"modified":"2024-03-09T12:14:49","modified_gmt":"2024-03-09T11:14:49","slug":"09-mar-24","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/09-mar-24\/","title":{"rendered":"La semana en Calculemus (9 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 e Isabelle\/HOL de las siguientes propiedades:<\/p>\n<ul>\n<li><a href=\"#ej1\">1. (s \\ t) \u222a (t \\ s) = (s \u222a t) \\ (s \u2229 t)<\/a><\/li>\n<li><a href=\"#ej2\">2. Pares \u222a Impares = Naturales<\/a><\/li>\n<li><a href=\"#ej3\">3. Los primos mayores que 2 son impares<\/a><\/li>\n<li><a href=\"#ej4\">4. s \u2229 (\u22c3 i, A i) = \u22c3 i, (A i \u2229 s)<\/a><\/li>\n<li><a href=\"#ej5\">5. (\u22c2 i, A i \u2229 B i) = (\u22c2 i, A i) \u2229 (\u22c2 i, B i)<\/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 (t \\ s) = (s \u222a t) \\ (s \u2229 t)<\/h3>\n<p>Demostrar con Lean4 que<br \/>\n&#92;[ (s &#92;setminus t) \u222a (t &#92;setminus s) = (s \u222a t) &#92;setminus (s \u2229 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 \\\\ t) \u222a (t \\\\ s) = (s \u222a t) \\\\ (s \u2229 t) :=\nby sorry\n<\/pre>\n<h2>1. Demostraci\u00f3n en lenguaje natural<\/h2>\n<p>Tenemos que demostrar que, para todo &#92;(x&#92;),<br \/>\n&#92;[ x \u2208 (s &#92;setminus t) \u222a (t &#92;setminus s) \u2194 x \u2208 (s \u222a t) &#92;setminus (s \u2229 t) &#92;]<br \/>\nSe demuestra mediante la siguiente cadena de equivalencias:<br \/>\n&#92;begin{align}<br \/>\n     &amp;x \u2208 (s &#92;setminus t) \u222a (t &#92;setminus s) &#92;&#92;<br \/>\n   \u2194 &amp;x \u2208 (s &#92;setminus t) \u2228 x \u2208 (t &#92;setminus s) &#92;&#92;<br \/>\n   \u2194 &amp;(x \u2208 s \u2227 x \u2209 t) \u2228 x \u2208 (t &#92;setminus s) &#92;&#92;<br \/>\n   \u2194 &amp;(x \u2208 s \u2228 x \u2208 (t &#92; s)) \u2227 (x \u2209 t \u2228 x \u2208 (t &#92;setminus s)) &#92;&#92;<br \/>\n   \u2194 &amp;(x \u2208 s \u2228 (x \u2208 t \u2227 x \u2209 s)) \u2227 (x \u2209 t \u2228 (x \u2208 t \u2227 x \u2209 s)) &#92;&#92;<br \/>\n   \u2194 &amp;((x \u2208 s \u2228 x \u2208 t) \u2227 (x \u2208 s \u2228 x \u2209 s)) \u2227 ((x \u2209 t \u2228 x \u2208 t) \u2227 (x \u2209 t \u2228 x \u2209 s)) &#92;&#92;<br \/>\n   \u2194 &amp;(x \u2208 s \u2228 x \u2208 t) \u2227 (x \u2209 t \u2228 x \u2209 s) &#92;&#92;<br \/>\n   \u2194 &amp;(x \u2208 s \u222a t) \u2227 (x \u2209 t \u2228 x \u2209 s) &#92;&#92;<br \/>\n   \u2194 &amp;(x \u2208 s \u222a t) \u2227 (x \u2209 s \u2228 x \u2209 t) &#92;&#92;<br \/>\n   \u2194 &amp;(x \u2208 s \u222a t) \u2227 \u00ac(x \u2208 s \u2227 x \u2208 t) &#92;&#92;<br \/>\n   \u2194 &amp;(x \u2208 s \u222a t) \u2227 \u00ac(x \u2208 s \u2229 t) &#92;&#92;<br \/>\n   \u2194 &amp;x \u2208 (s \u222a t) &#92;setminus (s \u2229 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) = (s \u222a t) \\ (s \u2229 t) :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 (s \\ t) \u222a (t \\ s) \u2194 x \u2208 (s \u222a t) \\ (s \u2229 t)\n  calc x \u2208 (s \\ t) \u222a (t \\ s)\n     \u2194 x \u2208 (s \\ t) \u2228 x \u2208 (t \\ s) :=\n         by exact mem_union x (s \\ t) (t \\ s)\n   _ \u2194 (x \u2208 s \u2227 x \u2209 t) \u2228 x \u2208 (t \\ s) :=\n         by simp only [mem_diff]\n   _ \u2194 (x \u2208 s \u2228 x \u2208 (t \\ s)) \u2227 (x \u2209 t \u2228 x \u2208 (t \\ s)) :=\n         by exact and_or_right\n   _ \u2194 (x \u2208 s \u2228 (x \u2208 t \u2227 x \u2209 s)) \u2227 (x \u2209 t \u2228 (x \u2208 t \u2227 x \u2209 s)) :=\n         by simp only [mem_diff]\n   _ \u2194 ((x \u2208 s \u2228 x \u2208 t) \u2227 (x \u2208 s \u2228 x \u2209 s)) \u2227\n       ((x \u2209 t \u2228 x \u2208 t) \u2227 (x \u2209 t \u2228 x \u2209 s)) :=\n         by simp_all only [or_and_left]\n   _ \u2194 ((x \u2208 s \u2228 x \u2208 t) \u2227 True) \u2227\n       (True \u2227 (x \u2209 t \u2228 x \u2209 s)) :=\n         by simp only [em (x \u2208 s), em' (x \u2208 t)]\n   _ \u2194 (x \u2208 s \u2228 x \u2208 t) \u2227 (x \u2209 t \u2228 x \u2209 s) :=\n         by simp only [and_true_iff (x \u2208 s \u2228 x \u2208 t),\n                       true_and_iff (\u00acx \u2208 t \u2228 \u00acx \u2208 s)]\n   _ \u2194 (x \u2208 s \u222a t) \u2227 (x \u2209 t \u2228 x \u2209 s) :=\n         by simp only [mem_union]\n   _ \u2194 (x \u2208 s \u222a t) \u2227 (x \u2209 s \u2228 x \u2209 t) :=\n         by simp only [or_comm]\n   _ \u2194 (x \u2208 s \u222a t) \u2227 \u00ac(x \u2208 s \u2227 x \u2208 t) :=\n         by simp only [not_and_or]\n   _ \u2194 (x \u2208 s \u222a t) \u2227 \u00ac(x \u2208 s \u2229 t) :=\n         by simp only [mem_inter_iff]\n   _ \u2194 x \u2208 (s \u222a t) \\ (s \u2229 t)     :=\n         by simp only [mem_diff]\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (s \\ t) \u222a (t \\ s) = (s \u222a t) \\ (s \u2229 t) :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 (s \\ t) \u222a (t \\ s) \u2194 x \u2208 (s \u222a t) \\ (s \u2229 t)\n  constructor\n  . -- \u22a2 x \u2208 (s \\ t) \u222a (t \\ s) \u2192 x \u2208 (s \u222a t) \\ (s \u2229 t)\n    rintro (\u27e8xs, xnt\u27e9 | \u27e8xt, xns\u27e9)\n    . -- xs : x \u2208 s\n      -- xnt : \u00acx \u2208 t\n      -- \u22a2 x \u2208 (s \u222a t) \\ (s \u2229 t)\n      constructor\n      . -- \u22a2 x \u2208 s \u222a t\n        left\n        -- \u22a2 x \u2208 s\n        exact xs\n      . -- \u22a2 \u00acx \u2208 s \u2229 t\n        rintro \u27e8-, xt\u27e9\n        -- xt : x \u2208 t\n        -- \u22a2 False\n        exact xnt xt\n    . -- xt : x \u2208 t\n      -- xns : \u00acx \u2208 s\n      -- \u22a2 x \u2208 (s \u222a t) \\ (s \u2229 t)\n      constructor\n      . -- \u22a2 x \u2208 s \u222a t\n        right\n        -- \u22a2 x \u2208 t\n        exact xt\n      . -- \u22a2 \u00acx \u2208 s \u2229 t\n        rintro \u27e8xs, -\u27e9\n        -- xs : x \u2208 s\n        -- \u22a2 False\n        exact xns xs\n  . -- \u22a2 x \u2208 (s \u222a t) \\ (s \u2229 t) \u2192 x \u2208 (s \\ t) \u222a (t \\ s)\n    rintro \u27e8xs | xt, nxst\u27e9\n    . -- xs : x \u2208 s\n      -- \u22a2 x \u2208 (s \\ t) \u222a (t \\ s)\n      left\n      -- \u22a2 x \u2208 s \\ t\n      use xs\n      -- \u22a2 \u00acx \u2208 t\n      intro xt\n      -- xt : x \u2208 t\n      -- \u22a2 False\n      apply nxst\n      -- \u22a2 x \u2208 s \u2229 t\n      constructor\n      . -- \u22a2 x \u2208 s\n        exact xs\n      . -- \u22a2 x \u2208 t\n        exact xt\n    . -- nxst : \u00acx \u2208 s \u2229 t\n      -- xt : x \u2208 t\n      -- \u22a2 x \u2208 (s \\ t) \u222a (t \\ s)\n      right\n      -- \u22a2 x \u2208 t \\ s\n      use xt\n      -- \u22a2 \u00acx \u2208 s\n      intro xs\n      -- xs : x \u2208 s\n      -- \u22a2 False\n      apply nxst\n      -- \u22a2 x \u2208 s \u2229 t\n      constructor\n      . -- \u22a2 x \u2208 s\n        exact xs\n      . -- \u22a2 x \u2208 t\n        exact xt\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (s \\ t) \u222a (t \\ s) = (s \u222a t) \\ (s \u2229 t) :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 (s \\ t) \u222a (t \\ s) \u2194 x \u2208 (s \u222a t) \\ (s \u2229 t)\n  constructor\n  . -- \u22a2 x \u2208 (s \\ t) \u222a (t \\ s) \u2192 x \u2208 (s \u222a t) \\ (s \u2229 t)\n    rintro (\u27e8xs, xnt\u27e9 | \u27e8xt, xns\u27e9)\n    . -- xt : x \u2208 t\n      -- xns : \u00acx \u2208 s\n      -- \u22a2 x \u2208 (s \u222a t) \\ (s \u2229 t)\n      aesop\n    . -- xt : x \u2208 t\n      -- xns : \u00acx \u2208 s\n      -- \u22a2 x \u2208 (s \u222a t) \\ (s \u2229 t)\n      aesop\n  . rintro \u27e8xs | xt, nxst\u27e9\n    . -- xs : x \u2208 s\n      -- \u22a2 x \u2208 (s \\ t) \u222a (t \\ s)\n      aesop\n    . -- nxst : \u00acx \u2208 s \u2229 t\n      -- xt : x \u2208 t\n      -- \u22a2 x \u2208 (s \\ t) \u222a (t \\ s)\n      aesop\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (s \\ t) \u222a (t \\ s) = (s \u222a t) \\ (s \u2229 t) :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 (s \\ t) \u222a (t \\ s) \u2194 x \u2208 (s \u222a t) \\ (s \u2229 t)\n  constructor\n  . -- \u22a2 x \u2208 (s \\ t) \u222a (t \\ s) \u2192 x \u2208 (s \u222a t) \\ (s \u2229 t)\n    rintro (\u27e8xs, xnt\u27e9 | \u27e8xt, xns\u27e9) <;> aesop\n  . -- \u22a2 x \u2208 (s \u222a t) \\ (s \u2229 t) \u2192 x \u2208 (s \\ t) \u222a (t \\ s)\n    rintro \u27e8xs | xt, nxst\u27e9 <;> aesop\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (s \\ t) \u222a (t \\ s) = (s \u222a t) \\ (s \u2229 t) :=\nby\n  ext\n  constructor\n  . aesop\n  . aesop\n\n-- 6\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (s \\ t) \u222a (t \\ s) = (s \u222a t) \\ (s \u2229 t) :=\nby\n  ext\n  constructor <;> aesop\n\n-- 7\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (s \\ t) \u222a (t \\ s) = (s \u222a t) \\ (s \u2229 t) :=\nby\n  rw [ext_iff]\n  -- \u22a2 \u2200 (x : \u03b1), x \u2208 (s \\ t) \u222a (t \\ s) \u2194 x \u2208 (s \u222a t) \\ (s \u2229 t)\n  intro\n  -- x : \u03b1\n  -- \u22a2 x \u2208 (s \\ t) \u222a (t \\ s) \u2194 x \u2208 (s \u222a t) \\ (s \u2229 t)\n  rw [iff_def]\n  -- \u22a2 (x \u2208 (s \\ t) \u222a (t \\ s) \u2192 x \u2208 (s \u222a t) \\ (s \u2229 t)) \u2227\n  --   (x \u2208 (s \u222a t) \\ (s \u2229 t) \u2192 x \u2208 (s \\ t) \u222a (t \\ s))\n  aesop\n\n-- Lemas usados\n-- ============\n\n-- variable (x : \u03b1)\n-- variable (a b c : Prop)\n-- #check (mem_union x s t : x \u2208 s \u222a t \u2194 x \u2208 s \u2228 x \u2208 t)\n-- #check (mem_diff x : x \u2208 s \\ t \u2194 x \u2208 s \u2227 \u00acx \u2208 t)\n-- #check (and_or_right : (a \u2227 b) \u2228 c \u2194 (a \u2228 c) \u2227 (b \u2228 c))\n-- #check (or_and_left : a \u2228 (b \u2227 c) \u2194 (a \u2228 b) \u2227 (a \u2228 c))\n-- #check (em a : a \u2228 \u00ac a)\n-- #check (em' a : \u00ac a \u2228 a)\n-- #check (and_true_iff a : a \u2227 True \u2194 a)\n-- #check (true_and_iff a : True \u2227 a \u2194 a)\n-- #check (or_comm : a \u2228 b \u2194 b \u2228 a)\n-- #check (not_and_or : \u00ac(a \u2227 b) \u2194 \u00aca \u2228 \u00acb)\n-- #check (mem_inter_iff x s t : x \u2208 s \u2229 t \u2194 x \u2208 s \u2227 x \u2208 t)\n-- #check (ext_iff : s = t \u2194 \u2200 (x : \u03b1), x \u2208 s \u2194 x \u2208 t)\n-- #check (iff_def : (a \u2194 b) \u2194 (a \u2192 b) \u2227 (b \u2192 a))\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_union_e_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 Diferencia_de_union_e_interseccion\nimports Main\nbegin\n\n(* 1 demostraci\u00f3n *)\nlemma \"(s - t) \u222a (t - s) = (s \u222a t) - (s \u2229 t)\"\nproof (rule equalityI)\n  show \"(s - t) \u222a (t - s) \u2286 (s \u222a t) - (s \u2229 t)\"\n  proof (rule subsetI)\n    fix x\n    assume \"x \u2208 (s - t) \u222a (t - s)\"\n    then show \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n    proof (rule UnE)\n      assume \"x \u2208 s - t\"\n      then show \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n      proof (rule DiffE)\n        assume \"x \u2208 s\"\n        assume \"x \u2209 t\"\n        have \"x \u2208 s \u222a t\"\n          using \u2039x \u2208 s\u203a by (simp only: UnI1)\n        moreover\n        have \"x \u2209 s \u2229 t\"\n        proof (rule notI)\n          assume \"x \u2208 s \u2229 t\"\n          then have \"x \u2208 t\"\n            by (simp only: IntD2)\n          with \u2039x \u2209 t\u203a show False\n            by (rule notE)\n        qed\n        ultimately show \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n          by (rule DiffI)\n      qed\n    next\n      assume \"x \u2208 t - s\"\n      then show \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n      proof (rule DiffE)\n        assume \"x \u2208 t\"\n        assume \"x \u2209 s\"\n        have \"x \u2208 s \u222a t\"\n          using \u2039x \u2208 t\u203a by (simp only: UnI2)\n        moreover\n        have \"x \u2209 s \u2229 t\"\n        proof (rule notI)\n          assume \"x \u2208 s \u2229 t\"\n          then have \"x \u2208 s\"\n            by (simp only: IntD1)\n          with \u2039x \u2209 s\u203a show False\n            by (rule notE)\n        qed\n        ultimately show \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n          by (rule DiffI)\n      qed\n    qed\n  qed\nnext\n  show \"(s \u222a t) - (s \u2229 t) \u2286 (s - t) \u222a (t - s)\"\n  proof (rule subsetI)\n    fix x\n    assume \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n    then show \"x \u2208 (s - t) \u222a (t - s)\"\n    proof (rule DiffE)\n      assume \"x \u2208 s \u222a t\"\n      assume \"x \u2209 s \u2229 t\"\n      note \u2039x \u2208 s \u222a t\u203a\n      then show \"x \u2208 (s - t) \u222a (t - s)\"\n      proof (rule UnE)\n        assume \"x \u2208 s\"\n        have \"x \u2209 t\"\n        proof (rule notI)\n          assume \"x \u2208 t\"\n          with \u2039x \u2208 s\u203a have \"x \u2208 s \u2229 t\"\n            by (rule IntI)\n          with \u2039x \u2209 s \u2229 t\u203a show False\n            by (rule notE)\n        qed\n        with \u2039x \u2208 s\u203a have \"x \u2208 s - t\"\n          by (rule DiffI)\n        then show \"x \u2208 (s - t) \u222a (t - s)\"\n          by (simp only: UnI1)\n      next\n        assume \"x \u2208 t\"\n        have \"x \u2209 s\"\n        proof (rule notI)\n          assume \"x \u2208 s\"\n          then have \"x \u2208 s \u2229 t\"\n            using \u2039x \u2208 t\u203a by (rule IntI)\n          with \u2039x \u2209 s \u2229 t\u203a show False\n            by (rule notE)\n        qed\n        with \u2039x \u2208 t\u203a have \"x \u2208 t - s\"\n          by (rule DiffI)\n        then show \"x \u2208 (s - t) \u222a (t - s)\"\n          by (rule UnI2)\n      qed\n    qed\n  qed\nqed\n\n(* 2 demostraci\u00f3n *)\nlemma \"(s - t) \u222a (t - s) = (s \u222a t) - (s \u2229 t)\"\nproof\n  show \"(s - t) \u222a (t - s) \u2286 (s \u222a t) - (s \u2229 t)\"\n  proof\n    fix x\n    assume \"x \u2208 (s - t) \u222a (t - s)\"\n    then show \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n    proof\n      assume \"x \u2208 s - t\"\n      then show \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n      proof\n        assume \"x \u2208 s\"\n        assume \"x \u2209 t\"\n        have \"x \u2208 s \u222a t\"\n          using \u2039x \u2208 s\u203a by simp\n        moreover\n        have \"x \u2209 s \u2229 t\"\n        proof\n          assume \"x \u2208 s \u2229 t\"\n          then have \"x \u2208 t\"\n            by simp\n          with \u2039x \u2209 t\u203a show False\n            by simp\n        qed\n        ultimately show \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n          by simp\n      qed\n    next\n      assume \"x \u2208 t - s\"\n      then show \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n      proof\n        assume \"x \u2208 t\"\n        assume \"x \u2209 s\"\n        have \"x \u2208 s \u222a t\"\n          using \u2039x \u2208 t\u203a by simp\n        moreover\n        have \"x \u2209 s \u2229 t\"\n        proof\n          assume \"x \u2208 s \u2229 t\"\n          then have \"x \u2208 s\"\n            by simp\n          with \u2039x \u2209 s\u203a show False\n            by simp\n        qed\n        ultimately show \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n          by simp\n      qed\n    qed\n  qed\nnext\n  show \"(s \u222a t) - (s \u2229 t) \u2286 (s - t) \u222a (t - s)\"\n  proof\n    fix x\n    assume \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n    then show \"x \u2208 (s - t) \u222a (t - s)\"\n    proof\n      assume \"x \u2208 s \u222a t\"\n      assume \"x \u2209 s \u2229 t\"\n      note \u2039x \u2208 s \u222a t\u203a\n      then show \"x \u2208 (s - t) \u222a (t - s)\"\n      proof\n        assume \"x \u2208 s\"\n        have \"x \u2209 t\"\n        proof\n          assume \"x \u2208 t\"\n          with \u2039x \u2208 s\u203a have \"x \u2208 s \u2229 t\"\n            by simp\n          with \u2039x \u2209 s \u2229 t\u203a show False\n            by simp\n        qed\n        with \u2039x \u2208 s\u203a have \"x \u2208 s - t\"\n          by simp\n        then show \"x \u2208 (s - t) \u222a (t - s)\"\n          by simp\n      next\n        assume \"x \u2208 t\"\n        have \"x \u2209 s\"\n        proof\n          assume \"x \u2208 s\"\n          then have \"x \u2208 s \u2229 t\"\n            using \u2039x \u2208 t\u203a by simp\n          with \u2039x \u2209 s \u2229 t\u203a show False\n            by simp\n        qed\n        with \u2039x \u2208 t\u203a have \"x \u2208 t - s\"\n          by simp\n        then show \"x \u2208 (s - t) \u222a (t - s)\"\n          by simp\n      qed\n    qed\n  qed\nqed\n\n(* 3\u00aa demostraci\u00f3n *)\nlemma \"(s - t) \u222a (t - s) = (s \u222a t) - (s \u2229 t)\"\nproof\n  show \"(s - t) \u222a (t - s) \u2286 (s \u222a t) - (s \u2229 t)\"\n  proof\n    fix x\n    assume \"x \u2208 (s - t) \u222a (t - s)\"\n    then show \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n    proof\n      assume \"x \u2208 s - t\"\n      then show \"x \u2208 (s \u222a t) - (s \u2229 t)\" by simp\n    next\n      assume \"x \u2208 t - s\"\n      then show \"x \u2208 (s \u222a t) - (s \u2229 t)\" by simp\n    qed\n  qed\nnext\n  show \"(s \u222a t) - (s \u2229 t) \u2286 (s - t) \u222a (t - s)\"\n  proof\n    fix x\n    assume \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n    then show \"x \u2208 (s - t) \u222a (t - s)\"\n    proof\n      assume \"x \u2208 s \u222a t\"\n      assume \"x \u2209 s \u2229 t\"\n      note \u2039x \u2208 s \u222a t\u203a\n      then show \"x \u2208 (s - t) \u222a (t - s)\"\n      proof\n        assume \"x \u2208 s\"\n        then show \"x \u2208 (s - t) \u222a (t - s)\"\n          using \u2039x \u2209 s \u2229 t\u203a by simp\n      next\n        assume \"x \u2208 t\"\n        then show \"x \u2208 (s - t) \u222a (t - s)\"\n          using \u2039x \u2209 s \u2229 t\u203a by simp\n      qed\n    qed\n  qed\nqed\n\n(* 4\u00aa demostraci\u00f3n *)\n\nlemma \"(s - t) \u222a (t - s) = (s \u222a t) - (s \u2229 t)\"\nproof\n  show \"(s - t) \u222a (t - s) \u2286 (s \u222a t) - (s \u2229 t)\"\n  proof\n    fix x\n    assume \"x \u2208 (s - t) \u222a (t - s)\"\n    then show \"x \u2208 (s \u222a t) - (s \u2229 t)\" by auto\n  qed\nnext\n  show \"(s \u222a t) - (s \u2229 t) \u2286 (s - t) \u222a (t - s)\"\n  proof\n    fix x\n    assume \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n    then show \"x \u2208 (s - t) \u222a (t - s)\" by auto\n  qed\nqed\n\n(* 5\u00aa demostraci\u00f3n *)\n\nlemma \"(s - t) \u222a (t - s) = (s \u222a t) - (s \u2229 t)\"\nproof\n  show \"(s - t) \u222a (t - s) \u2286 (s \u222a t) - (s \u2229 t)\" by auto\nnext\n  show \"(s \u222a t) - (s \u2229 t) \u2286 (s - t) \u222a (t - s)\" by auto\nqed\n\n(* 6\u00aa demostraci\u00f3n *)\n\nlemma \"(s - t) \u222a (t - s) = (s \u222a t) - (s \u2229 t)\"\n  by auto\n\nend\n<\/pre>\n<p><a name=\"ej2\"><\/a><\/p>\n<h3>2. Pares \u222a Impares = Naturales<\/h3>\n<p>Los conjuntos de los n\u00fameros naturales, de los pares y de los impares se definen en Lean4 por<\/p>\n<pre lang=\"haskell\">\n   def Naturales : Set \u2115 := {n | True}\n   def Pares     : Set \u2115 := {n | Even n}\n   def Impares   : Set \u2115 := {n | \u00acEven n}\n<\/pre>\n<p>Demostrar con Lean4 que<\/p>\n<pre lang=\"haskell\">\n   Pares \u222a Impares = Naturales\n<\/pre>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Nat.Parity\nopen Set\n\ndef Naturales : Set \u2115 := {n | True}\ndef Pares     : Set \u2115 := {n | Even n}\ndef Impares   : Set \u2115 := {n | \u00acEven n}\n\nexample : Pares \u222a Impares = Naturales :=\nby sorry\n<\/pre>\n<h2>1. Demostraci\u00f3n en lenguaje natural<\/h2>\n<p>Tenemos que demostrar que<br \/>\n&#92;[ &#92;{n | &#92;text{Even}(n)&#92;} \u222a &#92;{n | \u00ac&#92;text{Even}(n)&#92;} = &#92;{n | &#92;text{True}&#92;} &#92;]<br \/>\nes decir,<br \/>\n&#92;[ n \u2208 &#92;{n | &#92;text{Even}(n)&#92;} \u222a &#92;{n | \u00ac&#92;text{Even}(n)&#92;} \u2194 n \u2208 &#92;{n | &#92;text{True}&#92;} &#92;]<br \/>\nque se reduce a<br \/>\n&#92;[ \u22a2 &#92;text{Even}(n) \u2228 \u00ac&#92;text{Even}(n) &#92;]<br \/>\nque es una tautolog\u00eda.<\/p>\n<h2>2. Demostraciones con Lean4<\/h2>\n<pre lang=\"lean\">\nimport Mathlib.Data.Nat.Parity\nopen Set\n\ndef Naturales : Set \u2115 := {n | True}\ndef Pares     : Set \u2115 := {n | Even n}\ndef Impares   : Set \u2115 := {n | \u00acEven n}\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : Pares \u222a Impares = Naturales :=\nby\n  unfold Pares Impares Naturales\n  -- \u22a2 {n | Even n} \u222a {n | \u00acEven n} = {n | True}\n  ext n\n  -- \u22a2 n \u2208 {n | Even n} \u222a {n | \u00acEven n} \u2194 n \u2208 {n | True}\n  simp\n  -- \u22a2 Even n \u2228 \u00acEven n\n  exact em (Even n)\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : Pares \u222a Impares = Naturales :=\nby\n  unfold Pares Impares Naturales\n  -- \u22a2 {n | Even n} \u222a {n | \u00acEven n} = {n | True}\n  ext n\n  -- \u22a2 n \u2208 {n | Even n} \u222a {n | \u00acEven n} \u2194 n \u2208 {n | True}\n  tauto\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_de_pares_e_impares.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_de_pares_e_impares\nimports Main\nbegin\n\ndefinition naturales :: \"nat set\" where\n  \"naturales = {n\u2208\u2115 . True}\"\n\ndefinition pares :: \"nat set\" where\n  \"pares = {n\u2208\u2115 . even n}\"\n\ndefinition impares :: \"nat set\" where\n  \"impares = {n\u2208\u2115 . \u00ac even n}\"\n\n(* 1\u00aa demostraci\u00f3n *)\nlemma \"pares \u222a impares = naturales\"\nproof -\n  have \"\u2200 n \u2208 \u2115 . even n \u2228 \u00ac even n \u27f7 True\"\n    by simp\n  then have \"{n \u2208 \u2115. even n} \u222a {n \u2208 \u2115. \u00ac even n} = {n \u2208 \u2115. True}\"\n    by auto\n  then show \"pares \u222a impares = naturales\"\n    by (simp add: naturales_def pares_def impares_def)\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\nlemma \"pares \u222a impares = naturales\"\n  unfolding naturales_def pares_def impares_def\n  by auto\n\nend\n<\/pre>\n<p><a name=\"ej3\"><\/a><\/p>\n<h3>3. Los primos mayores que 2 son impares<\/h3>\n<p>Los n\u00fameros primos, los mayores que 2 y los impares se definen en Lean4 por<\/p>\n<pre lang=\"lean\">\n   def Primos      : Set \u2115 := {n | Nat.Prime n}\n   def MayoresQue2 : Set \u2115 := {n | n > 2}\n   def Impares     : Set \u2115 := {n | \u00acEven n}\n<\/pre>\n<p>Demostrar con Lean4 que<\/p>\n<pre lang=\"lean\">\n   Primos \u2229 MayoresQue2 \u2286 Impares\n<\/pre>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Nat.Parity\nimport Mathlib.Data.Nat.Prime\nimport Mathlib.Tactic\n\nopen Nat\n\ndef Primos      : Set \u2115 := {n | Nat.Prime n}\ndef MayoresQue2 : Set \u2115 := {n | n > 2}\ndef Impares     : Set \u2115 := {n | \u00acEven n}\n\nexample : Primos \u2229 MayoresQue2 \u2286 Impares :=\nby sorry\n<\/pre>\n<h2>1. Demostraciones con Lean4<\/h2>\n<pre lang=\"lean\">\nimport Mathlib.Data.Nat.Parity\nimport Mathlib.Data.Nat.Prime\nimport Mathlib.Tactic\n\nopen Nat\n\ndef Primos      : Set \u2115 := {n | Nat.Prime n}\ndef MayoresQue2 : Set \u2115 := {n | n > 2}\ndef Impares     : Set \u2115 := {n | \u00acEven n}\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : Primos \u2229 MayoresQue2 \u2286 Impares :=\nby\n  unfold Primos MayoresQue2 Impares\n  -- \u22a2 {n | Nat.Prime n} \u2229 {n | n > 2} \u2286 {n | \u00acEven n}\n  intro n\n  -- n : \u2115\n  -- \u22a2 n \u2208 {n | Nat.Prime n} \u2229 {n | n > 2} \u2192 n \u2208 {n | \u00acEven n}\n  simp\n  -- \u22a2 Nat.Prime n \u2192 2 < n \u2192 \u00acEven n\n  intro hn\n  -- hn : Nat.Prime n\n  -- \u22a2 2 < n \u2192 \u00acEven n\n  rcases Prime.eq_two_or_odd hn with (h | h)\n  . -- h : n = 2\n    rw [h]\n    -- \u22a2 2 < 2 \u2192 \u00acEven 2\n    intro h1\n    -- h1 : 2 < 2\n    -- \u22a2 \u00acEven 2\n    exfalso\n    exact absurd h1 (lt_irrefl 2)\n  . -- h : n % 2 = 1\n    rw [even_iff]\n    -- \u22a2 2 < n \u2192 \u00acn % 2 = 0\n    rw [h]\n    -- \u22a2 2 < n \u2192 \u00ac1 = 0\n    intro\n    -- a : 2 < n\n    -- \u22a2 \u00ac1 = 0\n    exact one_ne_zero\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : Primos \u2229 MayoresQue2 \u2286 Impares :=\nby\n  unfold Primos MayoresQue2 Impares\n  -- \u22a2 {n | Nat.Prime n} \u2229 {n | n > 2} \u2286 {n | \u00acEven n}\n  rintro n \u27e8h1, h2\u27e9\n  -- n : \u2115\n  -- h1 : n \u2208 {n | Nat.Prime n}\n  -- h2 : n \u2208 {n | n > 2}\n  -- \u22a2 n \u2208 {n | \u00acEven n}\n  simp at *\n  -- h1 : Nat.Prime n\n  -- h2 : 2 < n\n  -- \u22a2 \u00acEven n\n  rcases Prime.eq_two_or_odd h1 with (h3 | h4)\n  . -- h3 : n = 2\n    rw [h3] at h2\n    -- h2 : 2 < 2\n    exfalso\n    -- \u22a2 False\n    exact absurd h2 (lt_irrefl 2)\n  . -- h4 : n % 2 = 1\n    rw [even_iff]\n    -- \u22a2 \u00acn % 2 = 0\n    rw [h4]\n    -- \u22a2 \u00ac1 = 0\n    exact one_ne_zero\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : Primos \u2229 MayoresQue2 \u2286 Impares :=\nby\n  unfold Primos MayoresQue2 Impares\n  -- \u22a2 {n | Nat.Prime n} \u2229 {n | n > 2} \u2286 {n | \u00acEven n}\n  rintro n \u27e8h1, h2\u27e9\n  -- n : \u2115\n  -- h1 : n \u2208 {n | Nat.Prime n}\n  -- h2 : n \u2208 {n | n > 2}\n  -- \u22a2 n \u2208 {n | \u00acEven n}\n  simp at *\n  -- h1 : Nat.Prime n\n  -- h2 : 2 < n\n  -- \u22a2 \u00acEven n\n  rcases Prime.eq_two_or_odd h1 with (h3 | h4)\n  . -- h3 : n = 2\n    rw [h3] at h2\n    -- h2 : 2 < 2\n    linarith\n  . -- h4 : n % 2 = 1\n    rw [even_iff]\n    -- \u22a2 \u00acn % 2 = 0\n    linarith\n\n-- Lemas usados\n-- ============\n\n-- variable (p n : \u2115)\n-- variable (a b : Prop)\n-- #check (Prime.eq_two_or_odd : Nat.Prime p \u2192 p = 2 \u2228 p % 2 = 1)\n-- #check (absurd : a \u2192 \u00aca \u2192 b)\n-- #check (even_iff : Even n \u2194 n % 2 = 0)\n-- #check (lt_irrefl n : \u00acn < n)\n-- #check (one_ne_zero : 1 \u2260 0)\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_de_los_primos_y_los_mayores_que_dos.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_de_los_primos_y_los_mayores_que_dos\nimports Main \"HOL-Number_Theory.Number_Theory\"\nbegin\n\ndefinition primos :: \"nat set\" where\n  \"primos = {n \u2208 \u2115 . prime n}\"\n\ndefinition mayoresQue2 :: \"nat set\" where\n  \"mayoresQue2 = {n \u2208 \u2115 . n > 2}\"\n\ndefinition impares :: \"nat set\" where\n  \"impares = {n \u2208 \u2115 . \u00ac even n}\"\n\n(* 1\u00aa demostraci\u00f3n *)\nlemma \"primos \u2229 mayoresQue2 \u2286 impares\"\nproof\n  fix x\n  assume \"x \u2208 primos \u2229 mayoresQue2\"\n  then have \"x \u2208 \u2115 \u2227 prime x \u2227 2 < x\"\n    by (simp add: primos_def mayoresQue2_def)\n  then have \"x \u2208 \u2115 \u2227 odd x\"\n    by (simp add: prime_odd_nat)\n  then show \"x \u2208 impares\"\n    by (simp add: impares_def)\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\nlemma \"primos \u2229 mayoresQue2 \u2286 impares\"\n  unfolding primos_def mayoresQue2_def impares_def\n  by (simp add: Collect_mono_iff Int_def prime_odd_nat)\n\n(* 3\u00aa demostraci\u00f3n *)\nlemma \"primos \u2229 mayoresQue2 \u2286 impares\"\n  unfolding primos_def mayoresQue2_def impares_def\n  by (auto simp add: prime_odd_nat)\n\nend\n<\/pre>\n<p><a name=\"ej4\"><\/a><\/p>\n<h3>4. s \u2229 (\u22c3 i, A i) = \u22c3 i, (A i \u2229 s)<\/h3>\n<p>Demostrar con Lean4 que<br \/>\n&#92;[ s \u2229 \u22c3_i A_i = \u22c3_i (A_i \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\nimport Mathlib.Data.Set.Lattice\nimport Mathlib.Tactic\n\nopen Set\n\nvariable {\u03b1 : Type}\nvariable (s : Set \u03b1)\nvariable (A : \u2115 \u2192 Set \u03b1)\n\nexample : s \u2229 (\u22c3 i, A i) = \u22c3 i, (A i \u2229 s) :=\nby sorry\n<\/pre>\n<h2>1. Demostraci\u00f3n en lenguaje natural<\/h2>\n<p>Tenemos que demostrar que para cada &#92;(x&#92;), se verifica que<br \/>\n&#92;[ x \u2208 s \u2229 \u22c3_i A_i \u2194 x \u2208 \u22c3_i A_i \u2229 s &#92;]<br \/>\nLo demostramos mediante la siguiente cadena de equivalencias<br \/>\n&#92;begin{align}<br \/>\n   x \u2208 s \u2229 \u22c3_i A_i &amp;\u2194 x \u2208 s \u2227 x \u2208 \u22c3_i A_i &#92;&#92;<br \/>\n                   &amp;\u2194 x \u2208 s \u2227 (\u2203 i)[x \u2208 A_i] &#92;&#92;<br \/>\n                   &amp;\u2194 (\u2203 i)[x \u2208 s \u2227 x \u2208 A_i] &#92;&#92;<br \/>\n                   &amp;\u2194 (\u2203 i)[x \u2208 A_i \u2227 x \u2208 s] &#92;&#92;<br \/>\n                   &amp;\u2194 (\u2203 i)[x \u2208 A_i \u2229 s] &#92;&#92;<br \/>\n                   &amp;\u2194 x \u2208 \u22c3_i (A i \u2229 s)<br \/>\n&#92;end{align}<\/p>\n<h2>2. Demostraciones con Lean4<\/h2>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Basic\nimport Mathlib.Data.Set.Lattice\nimport Mathlib.Tactic\n\nopen Set\n\nvariable {\u03b1 : Type}\nvariable (s : Set \u03b1)\nvariable (A : \u2115 \u2192 Set \u03b1)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (\u22c3 i, A i) = \u22c3 i, (A i \u2229 s) :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 s \u2229 \u22c3 (i : \u2115), A i \u2194 x \u2208 \u22c3 (i : \u2115), A i \u2229 s\n  calc x \u2208 s \u2229 \u22c3 (i : \u2115), A i\n     \u2194 x \u2208 s \u2227 x \u2208 \u22c3 (i : \u2115), A i :=\n         by simp only [mem_inter_iff]\n   _ \u2194 x \u2208 s \u2227 (\u2203 i : \u2115, x \u2208 A i) :=\n         by simp only [mem_iUnion]\n   _ \u2194 \u2203 i : \u2115, x \u2208 s \u2227 x \u2208 A i :=\n         by simp only [exists_and_left]\n   _ \u2194 \u2203 i : \u2115, x \u2208 A i \u2227 x \u2208 s :=\n         by simp only [and_comm]\n   _ \u2194 \u2203 i : \u2115, x \u2208 A i \u2229 s :=\n         by simp only [mem_inter_iff]\n   _ \u2194 x \u2208 \u22c3 (i : \u2115), A i \u2229 s :=\n         by simp only [mem_iUnion]\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (\u22c3 i, A i) = \u22c3 i, (A i \u2229 s) :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 s \u2229 \u22c3 (i : \u2115), A i \u2194 x \u2208 \u22c3 (i : \u2115), A i \u2229 s\n  constructor\n  . -- \u22a2 x \u2208 s \u2229 \u22c3 (i : \u2115), A i \u2192 x \u2208 \u22c3 (i : \u2115), A i \u2229 s\n    intro h\n    -- h : x \u2208 s \u2229 \u22c3 (i : \u2115), A i\n    -- \u22a2 x \u2208 \u22c3 (i : \u2115), A i \u2229 s\n    rw [mem_iUnion]\n    -- \u22a2 \u2203 i, x \u2208 A i \u2229 s\n    rcases h with \u27e8xs, xUAi\u27e9\n    -- xs : x \u2208 s\n    -- xUAi : x \u2208 \u22c3 (i : \u2115), A i\n    rw [mem_iUnion] at xUAi\n    -- xUAi : \u2203 i, x \u2208 A i\n    rcases xUAi with \u27e8i, xAi\u27e9\n    -- i : \u2115\n    -- xAi : x \u2208 A i\n    use i\n    -- \u22a2 x \u2208 A i \u2229 s\n    constructor\n    . -- \u22a2 x \u2208 A i\n      exact xAi\n    . -- \u22a2 x \u2208 s\n      exact xs\n  . -- \u22a2 x \u2208 \u22c3 (i : \u2115), A i \u2229 s \u2192 x \u2208 s \u2229 \u22c3 (i : \u2115), A i\n    intro h\n    -- h : x \u2208 \u22c3 (i : \u2115), A i \u2229 s\n    -- \u22a2 x \u2208 s \u2229 \u22c3 (i : \u2115), A i\n    rw [mem_iUnion] at h\n    -- h : \u2203 i, x \u2208 A i \u2229 s\n    rcases h with \u27e8i, hi\u27e9\n    -- i : \u2115\n    -- hi : x \u2208 A i \u2229 s\n    rcases hi with \u27e8xAi, xs\u27e9\n    -- xAi : x \u2208 A i\n    -- xs : x \u2208 s\n    constructor\n    . -- \u22a2 x \u2208 s\n      exact xs\n    . -- \u22a2 x \u2208 \u22c3 (i : \u2115), A i\n      rw [mem_iUnion]\n      -- \u22a2 \u2203 i, x \u2208 A i\n      use i\n      -- \u22a2 x \u2208 A i\n      exact xAi\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (\u22c3 i, A i) = \u22c3 i, (A i \u2229 s) :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 s \u2229 \u22c3 (i : \u2115), A i \u2194 x \u2208 \u22c3 (i : \u2115), A i \u2229 s\n  simp\n  -- \u22a2 (x \u2208 s \u2227 \u2203 i, x \u2208 A i) \u2194 (\u2203 i, x \u2208 A i) \u2227 x \u2208 s\n  constructor\n  . -- \u22a2 (x \u2208 s \u2227 \u2203 i, x \u2208 A i) \u2192 (\u2203 i, x \u2208 A i) \u2227 x \u2208 s\n    rintro \u27e8xs, \u27e8i, xAi\u27e9\u27e9\n    -- xs : x \u2208 s\n    -- i : \u2115\n    -- xAi : x \u2208 A i\n    -- \u22a2 (\u2203 i, x \u2208 A i) \u2227 x \u2208 s\n    exact \u27e8\u27e8i, xAi\u27e9, xs\u27e9\n  . -- \u22a2 (\u2203 i, x \u2208 A i) \u2227 x \u2208 s \u2192 x \u2208 s \u2227 \u2203 i, x \u2208 A i\n    rintro \u27e8\u27e8i, xAi\u27e9, xs\u27e9\n    -- xs : x \u2208 s\n    -- i : \u2115\n    -- xAi : x \u2208 A i\n    -- \u22a2 x \u2208 s \u2227 \u2203 i, x \u2208 A i\n    exact \u27e8xs, \u27e8i, xAi\u27e9\u27e9\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (\u22c3 i, A i) = \u22c3 i, (A i \u2229 s) :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 s \u2229 \u22c3 (i : \u2115), A i \u2194 x \u2208 \u22c3 (i : \u2115), A i \u2229 s\n  aesop\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (\u22c3 i, A i) = \u22c3 i, (A i \u2229 s) :=\nby ext; aesop\n\n-- Lemas usados\n-- ============\n\n-- variable (x : \u03b1)\n-- variable (t : Set \u03b1)\n-- variable (a b : Prop)\n-- variable (p : \u03b1 \u2192 Prop)\n-- #check (mem_iUnion : x \u2208 \u22c3 i, A i \u2194 \u2203 i, x \u2208 A i)\n-- #check (mem_inter_iff x s t : x \u2208 s \u2229 t \u2194 x \u2208 s \u2227 x \u2208 t)\n-- #check (exists_and_left : (\u2203 (x : \u03b1), b \u2227 p x) \u2194 b \u2227 \u2203 (x : \u03b1), p x)\n-- #check (and_comm : a \u2227 b \u2194 b \u2227 a)\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\/Distributiva_de_la_interseccion_respecto_de_la_union_general.lean\" rel=\"noopener noreferrer\" target=\"_blank\">Lean 4 Web<\/a>.<\/p>\n<h2>3. Demostraciones con Isabelle\/HOL<\/h2>\n<pre lang=\"isar\">\ntheory Distributiva_de_la_interseccion_respecto_de_la_union_general\nimports Main\nbegin\n\n(* 1\u00aa demostraci\u00f3n *)\nlemma \"s \u2229 (\u22c3 i \u2208 I. A i) = (\u22c3 i \u2208 I. (A i \u2229 s))\"\nproof (rule equalityI)\n  show \"s \u2229 (\u22c3 i \u2208 I. A i) \u2286 (\u22c3 i \u2208 I. (A i \u2229 s))\"\n  proof (rule subsetI)\n    fix x\n    assume \"x \u2208 s \u2229 (\u22c3 i \u2208 I. A i)\"\n    then have \"x \u2208 s\"\n      by (simp only: IntD1)\n    have \"x \u2208 (\u22c3 i \u2208 I. A i)\"\n      using \u2039x \u2208 s \u2229 (\u22c3 i \u2208 I. A i)\u203a by (simp only: IntD2)\n    then show \"x \u2208 (\u22c3 i \u2208 I. (A i \u2229 s))\"\n    proof (rule UN_E)\n      fix i\n      assume \"i \u2208 I\"\n      assume \"x \u2208 A i\"\n      then have \"x \u2208 A i \u2229 s\"\n        using \u2039x \u2208 s\u203a by (rule IntI)\n      with \u2039i \u2208 I\u203a show \"x \u2208 (\u22c3 i \u2208 I. (A i \u2229 s))\"\n        by (rule UN_I)\n    qed\n  qed\nnext\n  show \"(\u22c3 i \u2208 I. (A i \u2229 s)) \u2286 s \u2229 (\u22c3 i \u2208 I. A i)\"\n  proof (rule subsetI)\n    fix x\n    assume \"x \u2208 (\u22c3 i \u2208 I. A i \u2229 s)\"\n    then show \"x \u2208 s \u2229 (\u22c3 i \u2208 I. A i)\"\n    proof (rule UN_E)\n      fix i\n      assume \"i \u2208 I\"\n      assume \"x \u2208 A i \u2229 s\"\n      then have \"x \u2208 A i\"\n        by (rule IntD1)\n      have \"x \u2208 s\"\n        using \u2039x \u2208 A i \u2229 s\u203a by (rule IntD2)\n      moreover\n      have \"x \u2208 (\u22c3 i \u2208 I. A i)\"\n        using \u2039i \u2208 I\u203a \u2039x \u2208 A i\u203a by (rule UN_I)\n      ultimately show \"x \u2208 s \u2229 (\u22c3 i \u2208 I. A i)\"\n        by (rule IntI)\n    qed\n  qed\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\nlemma \"s \u2229 (\u22c3 i \u2208 I. A i) = (\u22c3 i \u2208 I. (A i \u2229 s))\"\nproof\n  show \"s \u2229 (\u22c3 i \u2208 I. A i) \u2286 (\u22c3 i \u2208 I. (A i \u2229 s))\"\n  proof\n    fix x\n    assume \"x \u2208 s \u2229 (\u22c3 i \u2208 I. A i)\"\n    then have \"x \u2208 s\"\n      by simp\n    have \"x \u2208 (\u22c3 i \u2208 I. A i)\"\n      using \u2039x \u2208 s \u2229 (\u22c3 i \u2208 I. A i)\u203a by simp\n    then show \"x \u2208 (\u22c3 i \u2208 I. (A i \u2229 s))\"\n    proof\n      fix i\n      assume \"i \u2208 I\"\n      assume \"x \u2208 A i\"\n      then have \"x \u2208 A i \u2229 s\"\n        using \u2039x \u2208 s\u203a by simp\n      with \u2039i \u2208 I\u203a show \"x \u2208 (\u22c3 i \u2208 I. (A i \u2229 s))\"\n        by (rule UN_I)\n    qed\n  qed\nnext\n  show \"(\u22c3 i \u2208 I. (A i \u2229 s)) \u2286 s \u2229 (\u22c3 i \u2208 I. A i)\"\n  proof\n    fix x\n    assume \"x \u2208 (\u22c3 i \u2208 I. A i \u2229 s)\"\n    then show \"x \u2208 s \u2229 (\u22c3 i \u2208 I. A i)\"\n    proof\n      fix i\n      assume \"i \u2208 I\"\n      assume \"x \u2208 A i \u2229 s\"\n      then have \"x \u2208 A i\"\n        by simp\n      have \"x \u2208 s\"\n        using \u2039x \u2208 A i \u2229 s\u203a by simp\n      moreover\n      have \"x \u2208 (\u22c3 i \u2208 I. A i)\"\n        using \u2039i \u2208 I\u203a \u2039x \u2208 A i\u203a by (rule UN_I)\n      ultimately show \"x \u2208 s \u2229 (\u22c3 i \u2208 I. A i)\"\n        by simp\n    qed\n  qed\nqed\n\n(* 3\u00aa demostraci\u00f3n *)\nlemma \"s \u2229 (\u22c3 i \u2208 I. A i) = (\u22c3 i \u2208 I. (A i \u2229 s))\"\n  by auto\n\nend\n<\/pre>\n<p><a name=\"ej5\"><\/a><\/p>\n<h3>5. (\u22c2 i, A i \u2229 B i) = (\u22c2 i, A i) \u2229 (\u22c2 i, B i)<\/h3>\n<p>Demostrar con Lean4 que<br \/>\n&#92;[ \u22c2_i (A_i \u2229 B_i) = (\u22c2_i A_i) \u2229 (\u22c2_i B_i) &#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\n\nopen Set\n\nvariable {\u03b1 : Type}\nvariable (A B : \u2115 \u2192 Set \u03b1)\n\nexample : (\u22c2 i, A i \u2229 B i) = (\u22c2 i, A i) \u2229 (\u22c2 i, B i) :=\nby sorry\n<\/pre>\n<h2>1. Demostraci\u00f3n en lenguaje natural<\/h2>\n<p>Tenemos que demostrar que para &#92;(x&#92;) se verifica<br \/>\n&#92;[ x \u2208 \u22c2_i (A_i \u2229 B_i) \u2194 x \u2208 (\u22c2_i A_i) \u2229 (\u22c2_i B_i) &#92;]<br \/>\nLo demostramos mediante la siguiente cadena de equivalencias<br \/>\n&#92;begin{align}<br \/>\n   x \u2208 \u22c2_i (A_i \u2229 B_i) &amp;\u2194 (\u2200 i)[x \u2208 A_i \u2229 B_i] &#92;&#92;<br \/>\n                       &amp;\u2194 (\u2200 i)[x \u2208 A_i \u2227 x \u2208 B_i] &#92;&#92;<br \/>\n                       &amp;\u2194 (\u2200 i)[x \u2208 A_i] \u2227 (\u2200 i)[x \u2208 B_i] &#92;&#92;<br \/>\n                       &amp;\u2194 x \u2208 (\u22c2_i A_i) \u2227 x \u2208 (\u22c2_i B_i) &#92;&#92;<br \/>\n                       &amp;\u2194 x \u2208 (\u22c2_i A_i) \u2229 (\u22c2_i B_i)<br \/>\n&#92;end{align}<\/p>\n<h2>2. Demostraciones con Lean4<\/h2>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Basic\nimport Mathlib.Tactic\n\nopen Set\n\nvariable {\u03b1 : Type}\nvariable (A B : \u2115 \u2192 Set \u03b1)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (\u22c2 i, A i \u2229 B i) = (\u22c2 i, A i) \u2229 (\u22c2 i, B i) :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 \u22c2 (i : \u2115), A i \u2229 B i \u2194 x \u2208 (\u22c2 (i : \u2115), A i) \u2229 \u22c2 (i : \u2115), B i\n  calc x \u2208 \u22c2 i, A i \u2229 B i\n     \u2194 \u2200 i, x \u2208 A i \u2229 B i :=\n         by exact mem_iInter\n   _ \u2194 \u2200 i, x \u2208 A i \u2227 x \u2208 B i :=\n         by simp only [mem_inter_iff]\n   _ \u2194 (\u2200 i, x \u2208 A i) \u2227 (\u2200 i, x \u2208 B i) :=\n         by exact forall_and\n   _ \u2194 x \u2208 (\u22c2 i, A i) \u2227 x \u2208 (\u22c2 i, B i) :=\n         by simp only [mem_iInter]\n   _ \u2194 x \u2208 (\u22c2 i, A i) \u2229 \u22c2 i, B i :=\n         by simp only [mem_inter_iff]\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (\u22c2 i, A i \u2229 B i) = (\u22c2 i, A i) \u2229 (\u22c2 i, B i) :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 \u22c2 (i : \u2115), A i \u2229 B i \u2194 x \u2208 (\u22c2 (i : \u2115), A i) \u2229 \u22c2 (i : \u2115), B i\n  simp only [mem_inter_iff, mem_iInter]\n  -- \u22a2 (\u2200 (i : \u2115), x \u2208 A i \u2227 x \u2208 B i) \u2194 (\u2200 (i : \u2115), x \u2208 A i) \u2227 \u2200 (i : \u2115), x \u2208 B i\n  constructor\n  . -- \u22a2 (\u2200 (i : \u2115), x \u2208 A i \u2227 x \u2208 B i) \u2192 (\u2200 (i : \u2115), x \u2208 A i) \u2227 \u2200 (i : \u2115), x \u2208 B i\n    intro h\n    -- h : \u2200 (i : \u2115), x \u2208 A i \u2227 x \u2208 B i\n    -- \u22a2 (\u2200 (i : \u2115), x \u2208 A i) \u2227 \u2200 (i : \u2115), x \u2208 B i\n    constructor\n    . -- \u22a2 \u2200 (i : \u2115), x \u2208 A i\n      intro i\n      -- i : \u2115\n      -- \u22a2 x \u2208 A i\n      exact (h i).1\n    . -- \u22a2 \u2200 (i : \u2115), x \u2208 B i\n      intro i\n      -- i : \u2115\n      -- \u22a2 x \u2208 B i\n      exact (h i).2\n  . -- \u22a2 ((\u2200 (i : \u2115), x \u2208 A i) \u2227 \u2200 (i : \u2115), x \u2208 B i) \u2192 \u2200 (i : \u2115), x \u2208 A i \u2227 x \u2208 B i\n    intros h i\n    -- h : (\u2200 (i : \u2115), x \u2208 A i) \u2227 \u2200 (i : \u2115), x \u2208 B i\n    -- i : \u2115\n    -- \u22a2 x \u2208 A i \u2227 x \u2208 B i\n    rcases h with \u27e8h1, h2\u27e9\n    -- h1 : \u2200 (i : \u2115), x \u2208 A i\n    -- h2 : \u2200 (i : \u2115), x \u2208 B i\n    constructor\n    . -- \u22a2 x \u2208 A i\n      exact h1 i\n    . -- \u22a2 x \u2208 B i\n      exact h2 i\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (\u22c2 i, A i \u2229 B i) = (\u22c2 i, A i) \u2229 (\u22c2 i, B i) :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 \u22c2 (i : \u2115), A i \u2229 B i \u2194 x \u2208 (\u22c2 (i : \u2115), A i) \u2229 \u22c2 (i : \u2115), B i\n  simp only [mem_inter_iff, mem_iInter]\n  -- \u22a2 (\u2200 (i : \u2115), x \u2208 A i \u2227 x \u2208 B i) \u2194 (\u2200 (i : \u2115), x \u2208 A i) \u2227 \u2200 (i : \u2115), x \u2208 B i\n  exact \u27e8fun h \u21a6 \u27e8fun i \u21a6 (h i).1, fun i \u21a6 (h i).2\u27e9,\n         fun \u27e8h1, h2\u27e9 i \u21a6 \u27e8h1 i, h2 i\u27e9\u27e9\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (\u22c2 i, A i \u2229 B i) = (\u22c2 i, A i) \u2229 (\u22c2 i, B i) :=\nby\n  ext\n  -- x : \u03b1\n  -- \u22a2 x \u2208 \u22c2 (i : \u2115), A i \u2229 B i \u2194 x \u2208 (\u22c2 (i : \u2115), A i) \u2229 \u22c2 (i : \u2115), B i\n  simp only [mem_inter_iff, mem_iInter]\n  -- \u22a2 (\u2200 (i : \u2115), x \u2208 A i \u2227 x \u2208 B i) \u2194 (\u2200 (i : \u2115), x \u2208 A i) \u2227 \u2200 (i : \u2115), x \u2208 B i\n  aesop\n\n-- Lemas usados\n-- ============\n\n-- variable (x : \u03b1)\n-- variable (a b : Set \u03b1)\n-- variable (\u03b9 : Sort v)\n-- variable (s : \u03b9 \u2192 Set \u03b1)\n-- variable (p q : \u03b1 \u2192 Prop)\n-- #check (forall_and : (\u2200 (x : \u03b1), p x \u2227 q x) \u2194 (\u2200 (x : \u03b1), p x) \u2227 \u2200 (x : \u03b1), q x)\n-- #check (mem_iInter : x \u2208 \u22c2 (i : \u03b9), s i \u2194 \u2200 (i : \u03b9), x \u2208 s i)\n-- #check (mem_inter_iff x a b : x \u2208 a \u2229 b \u2194 x \u2208 a \u2227 x \u2208 b)\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_de_intersecciones.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_de_intersecciones\nimports Main\nbegin\n\n(* 1\u00aa demostraci\u00f3n *)\nlemma \"(\u22c2 i \u2208 I. A i \u2229 B i) = (\u22c2 i \u2208 I. A i) \u2229 (\u22c2 i \u2208 I. B i)\"\nproof (rule equalityI)\n  show \"(\u22c2 i \u2208 I. A i \u2229 B i) \u2286 (\u22c2 i \u2208 I. A i) \u2229 (\u22c2 i \u2208 I. B i)\"\n  proof (rule subsetI)\n    fix x\n    assume h1 : \"x \u2208 (\u22c2 i \u2208 I. A i \u2229 B i)\"\n    have \"x \u2208 (\u22c2 i \u2208 I. A i)\"\n    proof (rule INT_I)\n      fix i\n      assume \"i \u2208 I\"\n      with h1 have \"x \u2208 A i \u2229 B i\"\n        by (rule INT_D)\n      then show \"x \u2208 A i\"\n        by (rule IntD1)\n    qed\n    moreover\n    have \"x \u2208 (\u22c2 i \u2208 I. B i)\"\n    proof (rule INT_I)\n      fix i\n      assume \"i \u2208 I\"\n      with h1 have \"x \u2208 A i \u2229 B i\"\n        by (rule INT_D)\n      then show \"x \u2208 B i\"\n        by (rule IntD2)\n    qed\n    ultimately show \"x \u2208 (\u22c2 i \u2208 I. A i) \u2229 (\u22c2 i \u2208 I. B i)\"\n      by (rule IntI)\n  qed\nnext\n  show \"(\u22c2 i \u2208 I. A i) \u2229 (\u22c2 i \u2208 I. B i) \u2286 (\u22c2 i \u2208 I. A i \u2229 B i)\"\n  proof (rule subsetI)\n    fix x\n    assume h2 : \"x \u2208 (\u22c2 i \u2208 I. A i) \u2229 (\u22c2 i \u2208 I. B i)\"\n    show \"x \u2208 (\u22c2 i \u2208 I. A i \u2229 B i)\"\n    proof (rule INT_I)\n      fix i\n      assume \"i \u2208 I\"\n      have \"x \u2208 A i\"\n      proof -\n        have \"x \u2208 (\u22c2 i \u2208 I. A i)\"\n          using h2 by (rule IntD1)\n        then show \"x \u2208 A i\"\n          using \u2039i \u2208 I\u203a by (rule INT_D)\n      qed\n      moreover\n      have \"x \u2208 B i\"\n      proof -\n        have \"x \u2208 (\u22c2 i \u2208 I. B i)\"\n          using h2 by (rule IntD2)\n        then show \"x \u2208 B i\"\n          using \u2039i \u2208 I\u203a by (rule INT_D)\n      qed\n      ultimately show \"x \u2208 A i \u2229 B i\"\n        by (rule IntI)\n    qed\n  qed\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\nlemma \"(\u22c2 i \u2208 I. A i \u2229 B i) = (\u22c2 i \u2208 I. A i) \u2229 (\u22c2 i \u2208 I. B i)\"\nproof\n  show \"(\u22c2 i \u2208 I. A i \u2229 B i) \u2286 (\u22c2 i \u2208 I. A i) \u2229 (\u22c2 i \u2208 I. B i)\"\n  proof\n    fix x\n    assume h1 : \"x \u2208 (\u22c2 i \u2208 I. A i \u2229 B i)\"\n    have \"x \u2208 (\u22c2 i \u2208 I. A i)\"\n    proof\n      fix i\n      assume \"i \u2208 I\"\n      then show \"x \u2208 A i\"\n        using h1 by simp\n    qed\n    moreover\n    have \"x \u2208 (\u22c2 i \u2208 I. B i)\"\n    proof\n      fix i\n      assume \"i \u2208 I\"\n      then show \"x \u2208 B i\"\n        using h1 by simp\n    qed\n    ultimately show \"x \u2208 (\u22c2 i \u2208 I. A i) \u2229 (\u22c2 i \u2208 I. B i)\"\n      by simp\n  qed\nnext\n  show \"(\u22c2 i \u2208 I. A i) \u2229 (\u22c2 i \u2208 I. B i) \u2286 (\u22c2 i \u2208 I. A i \u2229 B i)\"\n  proof\n    fix x\n    assume h2 : \"x \u2208 (\u22c2 i \u2208 I. A i) \u2229 (\u22c2 i \u2208 I. B i)\"\n    show \"x \u2208 (\u22c2 i \u2208 I. A i \u2229 B i)\"\n    proof\n      fix i\n      assume \"i \u2208 I\"\n      then have \"x \u2208 A i\"\n        using h2 by simp\n      moreover\n      have \"x \u2208 B i\"\n        using \u2039i \u2208 I\u203a h2 by simp\n      ultimately show \"x \u2208 A i \u2229 B i\"\n        by simp\n    qed\nqed\nqed\n\n(* 3\u00aa demostraci\u00f3n *)\nlemma \"(\u22c2 i \u2208 I. A i \u2229 B i) = (\u22c2 i \u2208 I. A i) \u2229 (\u22c2 i \u2208 I. B i)\"\n  by auto\n\nend\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Demostraciones con Lean4 e Isabelle\/HOL publicadas esta semana en Calculemus.<\/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\/8155"}],"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=8155"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/8155\/revisions"}],"predecessor-version":[{"id":8156,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/8155\/revisions\/8156"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/media?parent=8155"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/categories?post=8155"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/tags?post=8155"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}