{"id":2286,"date":"2024-02-27T06:00:52","date_gmt":"2024-02-27T04:00:52","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=2286"},"modified":"2024-02-25T15:20:42","modified_gmt":"2024-02-25T13:20:42","slug":"27-feb-24","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/27-feb-24\/","title":{"rendered":"s \u2229 t = t \u2229 s"},"content":{"rendered":"\n

Demostrar con Lean4 que
\n\[ s \u2229 t = t \u2229 s \]<\/p>\n

Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n

\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
<\/p>\n
1. Demostraci\u00f3n en lenguaje natural<\/h2>\n
Tenemos que demostrar que
\n\[ (\u2200 x)[x \u2208 s \u2229 t \u2194 x \u2208 t \u2229 s] \]
\nDemostratemos la equivalencia por la doble implicaci\u00f3n.<\/p>\n
Sea \(x \u2208 s \u2229 t\). Entonces, se tiene
\n\begin{align}
\n   &x \u2208 s \tag{1} \\
\n   &x \u2208 t \tag{2}
\n\end{align}
\nLuego \(x \u2208 t \u2229 s\) (por (2) y (1)).<\/p>\n
La segunda implicaci\u00f3n se demuestra an\u00e1logamente.<\/p>\n
2. Demostraciones con Lean4<\/h2>\n
\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
Se puede interactuar con las demostraciones anteriores en Lean 4 Web<\/a>.<\/p>\n
3. Demostraciones con Isabelle\/HOL<\/h2>\n
\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","protected":false},"excerpt":{"rendered":"
Demostrar con Lean4 que \[ s \u2229 t = t \u2229 s \] Para ello, completar la siguiente teor\u00eda de Lean4: import Mathlib.Data.Set.Basic open Set variable {\u03b1 : Type} variable (s t : Set \u03b1) example : s \u2229 t = t \u2229 s := by sorry<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","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,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[7],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/2286"}],"collection":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/comments?post=2286"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/2286\/revisions"}],"predecessor-version":[{"id":2287,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/2286\/revisions\/2287"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=2286"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=2286"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=2286"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}