{"id":2329,"date":"2024-03-12T06:00:27","date_gmt":"2024-03-12T04:00:27","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=2329"},"modified":"2024-03-09T15:27:19","modified_gmt":"2024-03-09T13:27:19","slug":"12-mar-24","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/12-mar-24\/","title":{"rendered":"f\u207b\u00b9[u \u2229 v] = f\u207b\u00b9[u] \u2229 f\u207b\u00b9[v]"},"content":{"rendered":"\n

En Lean, la imagen inversa de un conjunto s<\/code> (de elementos de tipo \u03b2<\/code>) por la funci\u00f3n f<\/code> (de tipo \u03b1 \u2192 \u03b2<\/code>) es el conjunto f \u207b\u00b9' s<\/code> de elementos x<\/code> (de tipo \u03b1<\/code>) tales que f x \u2208 s<\/code>.<\/p>\n

Demostrar con Lean4 que<\/p>\n

\n   f \u207b\u00b9' (u \u2229 v) = f \u207b\u00b9' u \u2229 f \u207b\u00b9' v\n<\/pre>\n
Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n
\nimport Mathlib.Data.Set.Function\nvariable {\u03b1 \u03b2 : Type _}\nvariable (f : \u03b1 \u2192 \u03b2)\nvariable (u v : Set \u03b2)\nopen Set\n\nexample : f \u207b\u00b9' (u \u2229 v) = f \u207b\u00b9' u \u2229 f \u207b\u00b9' v :=\nby sorry\n<\/pre>\n
<\/p>\n
1. Demostraci\u00f3n en lenguaje natural<\/h2>\n
Tenemos que demostrar que, para todo \(x\),
\n\[ x \u2208 f\u207b\u00b9[u \u2229 v] \u2194 x \u2208 f\u207b\u00b9[u] \u2229 f\u207b\u00b9[v] \]
\nLo haremos mediante la siguiente cadena de equivalencias
\n\begin{align}
\n   x \u2208 f\u207b\u00b9[u \u2229 v] &\u2194 f x \u2208 u \u2229 v \\
\n                  &\u2194 f x \u2208 u \u2227 f x \u2208 v \\
\n                  &\u2194 x \u2208 f\u207b\u00b9[u] \u2227 x \u2208 f\u207b\u00b9[v] \\
\n                  &\u2194 x \u2208 f\u207b\u00b9[u] \u2229 f\u207b\u00b9[v] \\
\n\end{align}<\/p>\n
2. Demostraciones con Lean4<\/h2>\n
\nimport Mathlib.Data.Set.Function\n\nvariable {\u03b1 \u03b2 : Type _}\nvariable (f : \u03b1 \u2192 \u03b2)\nvariable (u v : Set \u03b2)\n\nopen Set\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f \u207b\u00b9' (u \u2229 v) = f \u207b\u00b9' u \u2229 f \u207b\u00b9' v :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 f \u207b\u00b9' (u \u2229 v) \u2194 x \u2208 f \u207b\u00b9' u \u2229 f \u207b\u00b9' v\n  calc x \u2208 f \u207b\u00b9' (u \u2229 v)\n     \u2194 f x \u2208 u \u2229 v :=\n         by simp only [mem_preimage]\n   _ \u2194 f x \u2208 u \u2227 f x \u2208 v :=\n         by simp only [mem_inter_iff]\n   _ \u2194 x \u2208 f \u207b\u00b9' u \u2227 x \u2208 f \u207b\u00b9' v :=\n         by simp only [mem_preimage]\n   _ \u2194 x \u2208 f \u207b\u00b9' u \u2229 f \u207b\u00b9' v :=\n         by simp only [mem_inter_iff]\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f \u207b\u00b9' (u \u2229 v) = f \u207b\u00b9' u \u2229 f \u207b\u00b9' v :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 f \u207b\u00b9' (u \u2229 v) \u2194 x \u2208 f \u207b\u00b9' u \u2229 f \u207b\u00b9' v\n  constructor\n  . -- \u22a2 x \u2208 f \u207b\u00b9' (u \u2229 v) \u2192 x \u2208 f \u207b\u00b9' u \u2229 f \u207b\u00b9' v\n    intro h\n    -- h : x \u2208 f \u207b\u00b9' (u \u2229 v)\n    -- \u22a2 x \u2208 f \u207b\u00b9' u \u2229 f \u207b\u00b9' v\n    constructor\n    . -- \u22a2 x \u2208 f \u207b\u00b9' u\n      apply mem_preimage.mpr\n      -- \u22a2 f x \u2208 u\n      rw [mem_preimage] at h\n      -- h : f x \u2208 u \u2229 v\n      exact mem_of_mem_inter_left h\n    . -- \u22a2 x \u2208 f \u207b\u00b9' v\n      apply mem_preimage.mpr\n      -- \u22a2 f x \u2208 v\n      rw [mem_preimage] at h\n      -- h : f x \u2208 u \u2229 v\n      exact mem_of_mem_inter_right h\n  . -- \u22a2 x \u2208 f \u207b\u00b9' u \u2229 f \u207b\u00b9' v \u2192 x \u2208 f \u207b\u00b9' (u \u2229 v)\n    intro h\n    -- h : x \u2208 f \u207b\u00b9' u \u2229 f \u207b\u00b9' v\n    -- \u22a2 x \u2208 f \u207b\u00b9' (u \u2229 v)\n    apply mem_preimage.mpr\n    -- \u22a2 f x \u2208 u \u2229 v\n    constructor\n    . -- \u22a2 f x \u2208 u\n      apply mem_preimage.mp\n      -- \u22a2 x \u2208 f \u207b\u00b9' u\n      exact mem_of_mem_inter_left h\n    . -- \u22a2 f x \u2208 v\n      apply mem_preimage.mp\n      -- \u22a2 x \u2208 f \u207b\u00b9' v\n      exact mem_of_mem_inter_right h\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f \u207b\u00b9' (u \u2229 v) = f \u207b\u00b9' u \u2229 f \u207b\u00b9' v :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 f \u207b\u00b9' (u \u2229 v) \u2194 x \u2208 f \u207b\u00b9' u \u2229 f \u207b\u00b9' v\n  constructor\n  . -- \u22a2 x \u2208 f \u207b\u00b9' (u \u2229 v) \u2192 x \u2208 f \u207b\u00b9' u \u2229 f \u207b\u00b9' v\n    intro h\n    -- h : x \u2208 f \u207b\u00b9' (u \u2229 v)\n    -- \u22a2 x \u2208 f \u207b\u00b9' u \u2229 f \u207b\u00b9' v\n    constructor\n    . -- \u22a2 x \u2208 f \u207b\u00b9' u\n      simp at *\n      -- h : f x \u2208 u \u2227 f x \u2208 v\n      -- \u22a2 f x \u2208 u\n      exact h.1\n    . -- \u22a2 x \u2208 f \u207b\u00b9' v\n      simp at *\n      -- h : f x \u2208 u \u2227 f x \u2208 v\n      -- \u22a2 f x \u2208 v\n      exact h.2\n  . -- \u22a2 x \u2208 f \u207b\u00b9' u \u2229 f \u207b\u00b9' v \u2192 x \u2208 f \u207b\u00b9' (u \u2229 v)\n    intro h\n    -- h : x \u2208 f \u207b\u00b9' u \u2229 f \u207b\u00b9' v\n    -- \u22a2 x \u2208 f \u207b\u00b9' (u \u2229 v)\n    simp at *\n    -- h : f x \u2208 u \u2227 f x \u2208 v\n    -- \u22a2 f x \u2208 u \u2227 f x \u2208 v\n    exact h\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f \u207b\u00b9' (u \u2229 v) = f \u207b\u00b9' u \u2229 f \u207b\u00b9' v :=\nby aesop\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f \u207b\u00b9' (u \u2229 v) = f \u207b\u00b9' u \u2229 f \u207b\u00b9' v :=\npreimage_inter\n\n-- 6\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f \u207b\u00b9' (u \u2229 v) = f \u207b\u00b9' u \u2229 f \u207b\u00b9' v :=\nrfl\n\n-- Lemas usados\n-- ============\n\n-- variable (x : \u03b1)\n-- variable (s t : Set \u03b1)\n-- #check (mem_of_mem_inter_left : x \u2208 s \u2229 t \u2192 x \u2208 s)\n-- #check (mem_of_mem_inter_right : x \u2208 s \u2229 t \u2192 x \u2208 t)\n-- #check (mem_preimage : x \u2208 f \u207b\u00b9' u \u2194 f x \u2208 u)\n-- #check (preimage_inter : f \u207b\u00b9' (u \u2229 v) = f \u207b\u00b9' u \u2229 f \u207b\u00b9' v)\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 Imagen_inversa_de_la_interseccion\nimports Main\nbegin\n\n(* 1\u00aa demostraci\u00f3n *)\nlemma \"f -` (u \u2229 v) = f -` u \u2229 f -` v\"\nproof (rule equalityI)\n  show \"f -` (u \u2229 v) \u2286 f -` u \u2229 f -` v\"\n  proof (rule subsetI)\n    fix x\n    assume \"x \u2208 f -` (u \u2229 v)\"\n    then have h : \"f x \u2208 u \u2229 v\"\n      by (simp only: vimage_eq)\n    have \"x \u2208 f -` u\"\n    proof -\n      have \"f x \u2208 u\"\n        using h by (rule IntD1)\n      then show \"x \u2208 f -` u\"\n        by (rule vimageI2)\n    qed\n    moreover\n    have \"x \u2208 f -` v\"\n    proof -\n      have \"f x \u2208 v\"\n        using h by (rule IntD2)\n      then show \"x \u2208 f -` v\"\n        by (rule vimageI2)\n    qed\n    ultimately show \"x \u2208 f -` u \u2229 f -` v\"\n      by (rule IntI)\n  qed\nnext\n  show \"f -` u \u2229 f -` v \u2286 f -` (u \u2229 v)\"\n  proof (rule subsetI)\n    fix x\n    assume h2 : \"x \u2208 f -` u \u2229 f -` v\"\n    have \"f x \u2208 u\"\n    proof -\n      have \"x \u2208 f -` u\"\n        using h2 by (rule IntD1)\n      then show \"f x \u2208 u\"\n        by (rule vimageD)\n    qed\n    moreover\n    have \"f x \u2208 v\"\n    proof -\n      have \"x \u2208 f -` v\"\n        using h2 by (rule IntD2)\n      then show \"f x \u2208 v\"\n        by (rule vimageD)\n    qed\n    ultimately have \"f x \u2208 u \u2229 v\"\n      by (rule IntI)\n    then show \"x \u2208 f -` (u \u2229 v)\"\n      by (rule vimageI2)\n  qed\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\nlemma \"f -` (u \u2229 v) = f -` u \u2229 f -` v\"\nproof\n  show \"f -` (u \u2229 v) \u2286 f -` u \u2229 f -` v\"\n  proof\n    fix x\n    assume \"x \u2208 f -` (u \u2229 v)\"\n    then have h : \"f x \u2208 u \u2229 v\"\n      by simp\n    have \"x \u2208 f -` u\"\n    proof -\n      have \"f x \u2208 u\"\n        using h by simp\n      then show \"x \u2208 f -` u\"\n        by simp\n    qed\n    moreover\n    have \"x \u2208 f -` v\"\n    proof -\n      have \"f x \u2208 v\"\n        using h by simp\n      then show \"x \u2208 f -` v\"\n        by simp\n    qed\n    ultimately show \"x \u2208 f -` u \u2229 f -` v\"\n      by simp\n  qed\nnext\n  show \"f -` u \u2229 f -` v \u2286 f -` (u \u2229 v)\"\n  proof\n    fix x\n    assume h2 : \"x \u2208 f -` u \u2229 f -` v\"\n    have \"f x \u2208 u\"\n    proof -\n      have \"x \u2208 f -` u\"\n        using h2 by simp\n      then show \"f x \u2208 u\"\n        by simp\n    qed\n    moreover\n    have \"f x \u2208 v\"\n    proof -\n      have \"x \u2208 f -` v\"\n        using h2 by simp\n      then show \"f x \u2208 v\"\n        by simp\n    qed\n    ultimately have \"f x \u2208 u \u2229 v\"\n      by simp\n    then show \"x \u2208 f -` (u \u2229 v)\"\n      by simp\n  qed\nqed\n\n(* 3\u00aa demostraci\u00f3n *)\nlemma \"f -` (u \u2229 v) = f -` u \u2229 f -` v\"\nproof\n  show \"f -` (u \u2229 v) \u2286 f -` u \u2229 f -` v\"\n  proof\n    fix x\n    assume h1 : \"x \u2208 f -` (u \u2229 v)\"\n    have \"x \u2208 f -` u\" using h1 by simp\n    moreover\n    have \"x \u2208 f -` v\" using h1 by simp\n    ultimately show \"x \u2208 f -` u \u2229 f -` v\" by simp\n  qed\nnext\n  show \"f -` u \u2229 f -` v \u2286 f -` (u \u2229 v)\"\n  proof\n    fix x\n    assume h2 : \"x \u2208 f -` u \u2229 f -` v\"\n    have \"f x \u2208 u\" using h2 by simp\n    moreover\n    have \"f x \u2208 v\" using h2 by simp\n    ultimately have \"f x \u2208 u \u2229 v\" by simp\n    then show \"x \u2208 f -` (u \u2229 v)\" by simp\n  qed\nqed\n\n(* 4\u00aa demostraci\u00f3n *)\nlemma \"f -` (u \u2229 v) = f -` u \u2229 f -` v\"\n  by (simp only: vimage_Int)\n\n(* 5\u00aa demostraci\u00f3n *)\nlemma \"f -` (u \u2229 v) = f -` u \u2229 f -` v\"\n  by auto\n\nend\n<\/pre>\n","protected":false},"excerpt":{"rendered":"
Demostrar con Lean4 e Isabelle\/HOL que f\u207b\u00b9[u \u2229 v] = f\u207b\u00b9[u] \u2229 f\u207b\u00b9[v]<\/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":[17],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/2329"}],"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=2329"}],"version-history":[{"count":4,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/2329\/revisions"}],"predecessor-version":[{"id":2333,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/2329\/revisions\/2333"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=2329"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=2329"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=2329"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}