{"id":7732,"date":"2022-05-01T12:35:02","date_gmt":"2022-05-01T10:35:02","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/?p=7732"},"modified":"2022-05-01T12:37:42","modified_gmt":"2022-05-01T10:37:42","slug":"dao-la-semana-en-calculemus-del-25-al-30-de-abril","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/dao-la-semana-en-calculemus-del-25-al-30-de-abril\/","title":{"rendered":"DAO: La semana en Calculemus (del 25 al 30 de abril)"},"content":{"rendered":"<p>Esta semana he publicado en <a href=\"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/\">Calculemus<\/a> las soluciones de los siguientes problemas:<\/p>\n<ul>\n<li><a href=\"#ej1\">1. Intersecci\u00f3n con su uni\u00f3n<\/a><\/li>\n<li><a href=\"#ej2\">2. Distributiva de la intersecci\u00f3n respecto de la uni\u00f3n general<\/a><\/li>\n<li><a href=\"#ej3\">3. Imagen inversa de la intersecci\u00f3n<\/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. Intersecci\u00f3n con su uni\u00f3n<\/h3>\n<p>Demostrar que <code>s \u2229 (s \u222a t) = s<\/code><\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport data.set.basic\nopen set\n\nvariable {\u03b1 : Type}\nvariables s t : set \u03b1\n\nexample : s \u2229 (s \u222a t) = s :=\nsorry\n<\/pre>\n<p><!--more--><\/p>\n<p><strong>Soluciones con Lean<\/strong><\/p>\n<pre lang=\"lean\">\nimport data.set.basic\nimport tactic\nopen set\n\nvariable {\u03b1 : Type}\nvariables s t : set \u03b1\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (s \u222a t) = s :=\nbegin\n  ext x,\n  split,\n  { intros h,\n    dsimp at h,\n    exact h.1, },\n  { intro xs,\n    dsimp,\n    split,\n    { exact xs, },\n    { left,\n      exact xs, }},\nend\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (s \u222a t) = s :=\nbegin\n  ext x,\n  split,\n  { intros h,\n    exact h.1, },\n  { intro xs,\n    split,\n    { exact xs, },\n    { left,\n      exact xs, }},\nend\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (s \u222a t) = s :=\nbegin\n  ext x,\n  split,\n  { intros h,\n    exact h.1, },\n  { intro xs,\n    split,\n    { exact xs, },\n    { exact (or.inl xs), }},\nend\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (s \u222a t) = s :=\nbegin\n  ext,\n  exact \u27e8\u03bb h, h.1,\n         \u03bb xs, \u27e8xs, or.inl xs\u27e9\u27e9,\nend\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (s \u222a t) = s :=\nbegin\n  ext,\n  exact \u27e8and.left,\n         \u03bb xs, \u27e8xs, or.inl xs\u27e9\u27e9,\nend\n\n-- 6\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (s \u222a t) = s :=\nbegin\n  ext x,\n  split,\n  { rintros \u27e8xs, _\u27e9,\n    exact xs },\n  { intro xs,\n    use xs,\n    left,\n    exact xs },\nend\n\n-- 7\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (s \u222a t) = s :=\nbegin\n  apply subset_antisymm,\n  { rintros x \u27e8hxs,-\u27e9,\n    exact hxs, },\n  { intros x hxs,\n    exact \u27e8hxs, or.inl hxs\u27e9, },\nend\n\n-- 8\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (s \u222a t) = s :=\n-- by suggest\ninf_sup_self\n\n-- 9\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (s \u222a t) = s :=\n-- by hint\nby finish\n<\/pre>\n<p>El c\u00f3digo de las demostraciones se encuentra en <a href=\"https:\/\/github.com\/jaalonso\/Razonando-con-Lean\/blob\/main\/src\/Interseccion_con_su_union.lean\">GitHub<\/a> y puede ejecutarse con el <a href=\"https:\/\/leanprover-community.github.io\/lean-web-editor\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Razonando-con-Lean\/main\/src\/Interseccion_con_su_union.lean\">Lean Web editor<\/a>.<\/p>\n<p>La construcci\u00f3n de las demostraciones se muestra en el siguiente v\u00eddeo<\/p>\n<p><iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/pQ4z5NCE5fU\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<p><strong>Soluciones con Isabelle\/HOL<\/strong><\/p>\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=\"ej2\"><\/a><\/p>\n<h3>2. Distributiva de la intersecci\u00f3n respecto de la uni\u00f3n general<\/h3>\n<p>Demostrar que <code>s \u2229 (\u22c3 i, A i) = \u22c3 i, (A i \u2229 s)<\/code><\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport data.set.basic\nimport data.set.lattice\nimport 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) :=\nsorry\n<\/pre>\n<p><!--more--><\/p>\n<p><strong>Soluciones con Lean<\/strong><\/p>\n<pre lang=\"lean\">\nimport data.set.basic\nimport data.set.lattice\nimport 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) :=\nbegin\n  ext x,\n  split,\n  { intro h,\n    rw mem_Union,\n    cases h with xs xUAi,\n    rw mem_Union at xUAi,\n    cases xUAi with i xAi,\n    use i,\n    split,\n    { exact xAi, },\n    { exact xs, }},\n  { intro h,\n    rw mem_Union at h,\n    cases h with i hi,\n    cases hi with xAi xs,\n    split,\n    { exact xs, },\n    { rw mem_Union,\n      use i,\n      exact xAi, }},\nend\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (\u22c3 i, A i) = \u22c3 i, (A i \u2229 s) :=\nbegin\n  ext x,\n  simp,\n  split,\n  { rintros \u27e8xs, \u27e8i, xAi\u27e9\u27e9,\n    exact \u27e8\u27e8i, xAi\u27e9, xs\u27e9, },\n  { rintros \u27e8\u27e8i, xAi\u27e9, xs\u27e9,\n    exact \u27e8xs, \u27e8i, xAi\u27e9\u27e9 },\nend\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (\u22c3 i, A i) = \u22c3 i, (A i \u2229 s) :=\nbegin\n  ext x,\n  finish,\nend\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (\u22c3 i, A i) = \u22c3 i, (A i \u2229 s) :=\nby ext; finish\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (\u22c3 i, A i) = \u22c3 i, (A i \u2229 s) :=\nby finish [ext_iff]\n\n-- 6\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u2229 (\u22c3 i, A i) = \u22c3 i, (A i \u2229 s) :=\nby tidy\n<\/pre>\n<p>El c\u00f3digo de las demostraciones se encuentra en <a href=\"https:\/\/github.com\/jaalonso\/Razonando-con-Lean\/blob\/main\/src\/Distributiva_de_la_interseccion_respecto_de_la_union_general.lean\">GitHub<\/a> y puede ejecutarse con el <a href=\"https:\/\/leanprover-community.github.io\/lean-web-editor\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Razonando-con-Lean\/main\/src\/Distributiva_de_la_interseccion_respecto_de_la_union_general.lean\">Lean Web editor<\/a>.<\/p>\n<p>La construcci\u00f3n de las demostraciones se muestra en el siguiente v\u00eddeo<\/p>\n<p><iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/lYd2xT-G3ZY\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<p><strong>Soluciones con Isabelle\/HOL<\/strong><\/p>\n<pre lang=\"isar\">\ntheory Distributiva_de_la_interseccion_respecto_de_la_union_general\nimports Main\nbegin\n\nsection \u20391\u00aa demostraci\u00f3n\u203a\n\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\nsection \u20392\u00aa demostraci\u00f3n\u203a\n\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\nsection \u20393\u00aa demostraci\u00f3n\u203a\n\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=\"ej3\"><\/a><\/p>\n<h3>3. Imagen inversa de la intersecci\u00f3n<\/h3>\n<p>En Lean, la imagen inversa de un conjunto s (de elementos de tipo  por la funci\u00f3n f (de tipo \u03b1 \u2192 \u03b2) es el conjunto <code>f \u207b\u00b9' s<\/code> de elementos x (de tipo \u03b1) tales que <code>f x \u2208 s<\/code>.<\/p>\n<p>Demostrar que <code>f \u207b\u00b9' (u \u2229 v) = f \u207b\u00b9' u \u2229 f \u207b\u00b9' v<\/code><\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport data.set.basic\n\nopen set\n\nvariables {\u03b1 : Type*} {\u03b2 : Type*}\nvariable  f : \u03b1 \u2192 \u03b2\nvariables u v : set \u03b2\n\nexample : f \u207b\u00b9' (u \u2229 v) = f \u207b\u00b9' u \u2229 f \u207b\u00b9' v :=\nsorry\n<\/pre>\n<p><!--more--><\/p>\n<p><strong>Soluciones con Lean<\/strong><\/p>\n<pre lang=\"lean\">\nimport data.set.basic\n\nopen set\n\nvariables {\u03b1 : Type*} {\u03b2 : Type*}\nvariable  f : \u03b1 \u2192 \u03b2\nvariables u v : set \u03b2\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f \u207b\u00b9' (u \u2229 v) = f \u207b\u00b9' u \u2229 f \u207b\u00b9' v :=\nbegin\n  ext x,\n  split,\n  { intro h,\n    split,\n    { apply mem_preimage.mpr,\n      rw mem_preimage at h,\n      exact mem_of_mem_inter_left h, },\n    { apply mem_preimage.mpr,\n      rw mem_preimage at h,\n      exact mem_of_mem_inter_right h, }},\n  { intro h,\n    apply mem_preimage.mpr,\n    split,\n    { apply mem_preimage.mp,\n      exact mem_of_mem_inter_left h,},\n    { apply mem_preimage.mp,\n      exact mem_of_mem_inter_right h, }},\nend\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f \u207b\u00b9' (u \u2229 v) = f \u207b\u00b9' u \u2229 f \u207b\u00b9' v :=\nbegin\n  ext x,\n  split,\n  { intro h,\n    split,\n    { simp at *,\n      exact h.1, },\n    { simp at *,\n      exact h.2, }},\n  { intro h,\n    simp at *,\n    exact h, },\nend\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f \u207b\u00b9' (u \u2229 v) = f \u207b\u00b9' u \u2229 f \u207b\u00b9' v :=\n-- by hint\nby finish\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f \u207b\u00b9' (u \u2229 v) = f \u207b\u00b9' u \u2229 f \u207b\u00b9' v :=\n-- by library_search\npreimage_inter\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f \u207b\u00b9' (u \u2229 v) = f \u207b\u00b9' u \u2229 f \u207b\u00b9' v :=\nrfl\n<\/pre>\n<p>El c\u00f3digo de las demostraciones se encuentra en <a href=\"https:\/\/github.com\/jaalonso\/Razonando-con-Lean\/blob\/main\/src\/Imagen_inversa_de_la_interseccion.lean\">GitHub<\/a> y puede ejecutarse con el <a href=\"https:\/\/leanprover-community.github.io\/lean-web-editor\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Razonando-con-Lean\/main\/src\/Imagen_inversa_de_la_interseccion.lean\">Lean Web editor<\/a>.<\/p>\n<p>La construcci\u00f3n de las demostraciones se muestra en el siguiente v\u00eddeo<\/p>\n<p><iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/av4vNL8-AJA\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<p><strong>Soluciones con Isabelle\/HOL<\/strong><\/p>\n<pre lang=\"isar\">\ntheory Imagen_inversa_de_la_interseccion\nimports Main\nbegin\n\nsection \u20391\u00aa demostraci\u00f3n\u203a\n\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\nsection \u20392\u00aa demostraci\u00f3n\u203a\n\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\nsection \u20393\u00aa demostraci\u00f3n\u203a\n\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\nsection \u20394\u00aa demostraci\u00f3n\u203a\n\nlemma \"f -` (u \u2229 v) = f -` u \u2229 f -` v\"\n  by (simp only: vimage_Int)\n\nsection \u20395\u00aa demostraci\u00f3n\u203a\n\nlemma \"f -` (u \u2229 v) = f -` u \u2229 f -` v\"\n  by auto\n\nend\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Esta semana he publicado en Calculemus las soluciones de los siguientes problemas: 1. Intersecci\u00f3n con su uni\u00f3n 2. Distributiva de la intersecci\u00f3n respecto de la uni\u00f3n general 3. Imagen inversa de la intersecci\u00f3n A continuaci\u00f3n se muestran las soluciones.<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"footnotes":"","_jetpack_memberships_contains_paid_content":false},"categories":[335],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_likes_enabled":false,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7732"}],"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=7732"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7732\/revisions"}],"predecessor-version":[{"id":7735,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7732\/revisions\/7735"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/media?parent=7732"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/categories?post=7732"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/tags?post=7732"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}