{"id":8174,"date":"2024-04-21T18:04:28","date_gmt":"2024-04-21T16:04:28","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/?p=8174"},"modified":"2024-04-21T18:09:51","modified_gmt":"2024-04-21T16:09:51","slug":"21-abr-24","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/21-abr-24\/","title":{"rendered":"La semana en Calculemus (21 de abril de 2024)"},"content":{"rendered":"<p>Desde el 18 de marzo, 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. Si f es inyectiva, entonces f\u207b\u00b9(f(s)\u200b) \u2286 s<\/a><\/li>\n<li><a href=\"#ej2\">2. f(f\u207b\u00b9(u)) \u2286 u<\/a><\/li>\n<li><a href=\"#ej3\">3. Si f es suprayectiva, entonces u \u2286 f(f\u207b\u00b9(u))<\/a><\/li>\n<li><a href=\"#ej4\">4. Si s \u2286 t, entonces f(s) \u2286 f(t)<\/a><\/li>\n<li><a href=\"#ej5\">5. Si u \u2286 v, entonces f\u207b\u00b9(u) \u2286 f\u207b\u00b9(v)<\/a><\/li>\n<li><a href=\"#ej6\">6. f\u207b\u00b9(A \u222a B) = f\u207b\u00b9(A) \u222a f\u207b\u00b9(B)<\/a><\/li>\n<li><a href=\"#ej7\">7. f(s \u2229 t) \u2286 f(s) \u2229 f(t)<\/a><\/li>\n<li><a href=\"#ej8\">8. Si f es inyectiva, entonces f(s) \u2229 f(t) \u2286 f(s \u2229 t)<\/a><\/li>\n<li><a href=\"#ej9\">9. f(s) \\ f(t) \u2286 f(s \\ t)<\/a><\/li>\n<li><a href=\"#ej10\">10. f(s) \u2229 t = f(s \u2229 f\u207b\u00b9(t))<\/a><\/li>\n<\/ul>\n\n<p>A continuaci\u00f3n se muestran las soluciones.<br \/>\n<!--more--><br \/>\n<a name=\"ej1\"><\/a><\/p>\n<h3>1. Si f es inyectiva, entonces f\u207b\u00b9[f[s]\u200b] \u2286 s<\/h3>\n<p>Demostrar con Lean4 que si &#92;(f&#92;) es inyectiva, entonces &#92;(f\u207b\u00b9[f[s]\u200b] \u2286 s&#92;).<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Function\nopen Set Function\nvariable {\u03b1 \u03b2 : Type _}\nvariable (f : \u03b1 \u2192 \u03b2)\nvariable (s : Set \u03b1)\n\nexample\n  (h : Injective f)\n  : f \u207b\u00b9' (f '' s) \u2286 s :=\nby sorry\n<\/pre>\n<h4>1.1. Demostraci\u00f3n en lenguaje natural<\/h4>\n<p>Sea &#92;(x&#92;) tal que<br \/>\n&#92;[ x \u2208 f\u207b\u00b9[f[s]] &#92;]<br \/>\nEntonces,<br \/>\n&#92;[ f(x) \u2208 f[s] &#92;]<br \/>\ny, por tanto, existe un<br \/>\n&#92;[ y \u2208 s &#92;tag{1} &#92;]<br \/>\ntal que<br \/>\n&#92;[ f(y) = f(x) &#92;tag{2} &#92;]<br \/>\nDe (2), puesto que &#92;(f&#92;) es inyectiva, se tiene que<br \/>\n&#92;[ y = x &#92;tag{3} &#92;]<br \/>\nFinalmente, de (3) y (1), se tiene que<br \/>\n&#92;[ x \u2208 s &#92;]<br \/>\nque es lo que ten\u00edamos que demostrar.<\/p>\n<h4>1.2. Demostraciones con Lean4<\/h4>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Function\n\nopen Set Function\n\nvariable {\u03b1 \u03b2 : Type _}\nvariable (f : \u03b1 \u2192 \u03b2)\nvariable (s : Set \u03b1)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : Injective f)\n  : f \u207b\u00b9' (f '' s) \u2286 s :=\nby\n  intros x hx\n  -- x : \u03b1\n  -- hx : x \u2208 f \u207b\u00b9' (f '' s)\n  -- \u22a2 x \u2208 s\n  have h1 : f x \u2208 f '' s := mem_preimage.mp hx\n  have h2 : \u2203 y, y \u2208 s \u2227 f y = f x := (mem_image f s (f x)).mp h1\n  obtain \u27e8y, hy : y \u2208 s \u2227 f y = f x\u27e9 := h2\n  obtain \u27e8ys : y \u2208 s, fyx : f y = f x\u27e9 := hy\n  have h3 : y = x := h fyx\n  show x \u2208 s\n  exact h3 \u25b8 ys\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : Injective f)\n  : f \u207b\u00b9' (f '' s) \u2286 s :=\nby\n  intros x hx\n  -- x : \u03b1\n  -- hx : x \u2208 f \u207b\u00b9' (f '' s)\n  -- \u22a2 x \u2208 s\n  rw [mem_preimage] at hx\n  -- hx : f x \u2208 f '' s\n  rw [mem_image] at hx\n  -- hx : \u2203 x_1, x_1 \u2208 s \u2227 f x_1 = f x\n  rcases hx with \u27e8y, hy\u27e9\n  -- y : \u03b1\n  -- hy : y \u2208 s \u2227 f y = f x\n  rcases hy with \u27e8ys, fyx\u27e9\n  -- ys : y \u2208 s\n  -- fyx : f y = f x\n  unfold Injective at h\n  -- h : \u2200 \u2983a\u2081 a\u2082 : \u03b1\u2984, f a\u2081 = f a\u2082 \u2192 a\u2081 = a\u2082\n  have h1 : y = x := h fyx\n  rw [\u2190h1]\n  -- \u22a2 y \u2208 s\n  exact ys\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : Injective f)\n  : f \u207b\u00b9' (f '' s) \u2286 s :=\nby\n  intros x hx\n  -- x : \u03b1\n  -- hx : x \u2208 f \u207b\u00b9' (f '' s)\n  -- \u22a2 x \u2208 s\n  rw [mem_preimage] at hx\n  -- hx : f x \u2208 f '' s\n  rcases hx with \u27e8y, ys, fyx\u27e9\n  -- y : \u03b1\n  -- ys : y \u2208 s\n  -- fyx : f y = f x\n  rw [\u2190h fyx]\n  -- \u22a2 y \u2208 s\n  exact ys\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : Injective f)\n  : f \u207b\u00b9' (f '' s) \u2286 s :=\nby\n  rintro x \u27e8y, ys, hy\u27e9\n  -- x y : \u03b1\n  -- ys : y \u2208 s\n  -- hy : f y = f x\n  -- \u22a2 x \u2208 s\n  rw [\u2190h hy]\n  -- \u22a2 y \u2208 s\n  exact ys\n\n-- Lemas usados\n-- ============\n\n-- variable (x : \u03b1)\n-- variable (y : \u03b2)\n-- variable (t : Set \u03b2)\n-- #check (mem_image f s y : y \u2208 f '' s \u2194 \u2203 (x : \u03b1), x \u2208 s \u2227 f x = y)\n-- #check (mem_preimage : x \u2208 f \u207b\u00b9' t \u2194 f x \u2208 t)\n<\/pre>\n<p>Se puede interactuar con las demostraciones anteriores en <a href=\"https:\/\/live.lean-lang.org\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus2\/main\/src\/Imagen_inversa_de_la_imagen_de_aplicaciones_inyectivas.lean\" rel=\"noopener noreferrer\" target=\"_blank\">Lean 4 Web<\/a>.<\/p>\n<h4>1.3. Demostraciones con Isabelle\/HOL<\/h4>\n<pre lang=\"isar\">\ntheory Imagen_inversa_de_la_imagen_de_aplicaciones_inyectivas\nimports Main\nbegin\n\n(* 1\u00aa demostraci\u00f3n *)\nlemma\n  assumes \"inj f\"\n  shows \"f -` (f ` s) \u2286 s\"\nproof (rule subsetI)\n  fix x\n  assume \"x \u2208 f -` (f ` s)\"\n  then have \"f x \u2208 f ` s\"\n    by (rule vimageD)\n  then show \"x \u2208 s\"\n  proof (rule imageE)\n    fix y\n    assume \"f x = f y\"\n    assume \"y \u2208 s\"\n    have \"x = y\"\n      using \u2039inj f\u203a \u2039f x = f y\u203a by (rule injD)\n    then show \"x \u2208 s\"\n      using \u2039y \u2208 s\u203a  by (rule ssubst)\n  qed\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\nlemma\n  assumes \"inj f\"\n  shows \"f -` (f ` s) \u2286 s\"\nproof\n  fix x\n  assume \"x \u2208 f -` (f ` s)\"\n  then have \"f x \u2208 f ` s\"\n    by simp\n  then show \"x \u2208 s\"\n  proof\n    fix y\n    assume \"f x = f y\"\n    assume \"y \u2208 s\"\n    have \"x = y\"\n      using \u2039inj f\u203a \u2039f x = f y\u203a by (rule injD)\n    then show \"x \u2208 s\"\n      using \u2039y \u2208 s\u203a  by simp\n  qed\nqed\n\n(* 3\u00aa demostraci\u00f3n *)\nlemma\n  assumes \"inj f\"\n  shows \"f -` (f ` s) \u2286 s\"\n  using assms\n  unfolding inj_def\n  by auto\n\n(* 4\u00aa demostraci\u00f3n *)\nlemma\n  assumes \"inj f\"\n  shows \"f -` (f ` s) \u2286 s\"\n  using assms\n  by (simp only: inj_vimage_image_eq)\n\nend\n<\/pre>\n<p><a name=\"ej2\"><\/a><\/p>\n<h3>2. f[f\u207b\u00b9[u]] \u2286 u<\/h3>\n<p>Demostrar con Lean4 que<br \/>\n&#92;[ f[f\u207b\u00b9[u]] \u2286 u &#92;]<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Function\nopen Set\n\nvariable {\u03b1 \u03b2 : Type _}\nvariable (f : \u03b1 \u2192 \u03b2)\nvariable (u : Set \u03b2)\n\nexample : f '' (f\u207b\u00b9' u) \u2286 u :=\nby sorry\n<\/pre>\n<h4>2.1. Demostraci\u00f3n en lenguaje natural<\/h4>\n<p>Sea &#92;(y \u2208 f[f\u207b\u00b9[u]]&#92;). Entonces existe un &#92;(x&#92;) tal que<br \/>\n&#92;begin{align}<br \/>\n   &amp;x \u2208 f\u207b\u00b9[u] &#92;tag{1} &#92;&#92;<br \/>\n   &amp;f(x) = y   &#92;tag{2}<br \/>\n&#92;end{align}<br \/>\nPor (1),<br \/>\n&#92;[ f(x) \u2208 u &#92;]<br \/>\ny, por (2),<br \/>\n&#92;[ y \u2208 u &#92;]<\/p>\n<h4>2.2. Demostraciones con Lean4<\/h4>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Function\nopen Set\n\nvariable {\u03b1 \u03b2 : Type _}\nvariable (f : \u03b1 \u2192 \u03b2)\nvariable (u : Set \u03b2)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f '' (f\u207b\u00b9' u) \u2286 u :=\nby\n  intros y h\n  -- y : \u03b2\n  -- h : y \u2208 f '' (f \u207b\u00b9' u)\n  -- \u22a2 y \u2208 u\n  obtain \u27e8x : \u03b1, h1 : x \u2208 f \u207b\u00b9' u \u2227 f x = y\u27e9 := h\n  obtain \u27e8hx : x \u2208 f \u207b\u00b9' u, fxy : f x = y\u27e9 := h1\n  have h2 : f x \u2208 u := mem_preimage.mp hx\n  show y \u2208 u\n  exact fxy \u25b8 h2\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f '' (f\u207b\u00b9' u) \u2286 u :=\nby\n  intros y h\n  -- y : \u03b2\n  -- h : y \u2208 f '' (f \u207b\u00b9' u)\n  -- \u22a2 y \u2208 u\n  rcases h with \u27e8x, h2\u27e9\n  -- x : \u03b1\n  -- h2 : x \u2208 f \u207b\u00b9' u \u2227 f x = y\n  rcases h2 with \u27e8hx, fxy\u27e9\n  -- hx : x \u2208 f \u207b\u00b9' u\n  -- fxy : f x = y\n  rw [\u2190fxy]\n  -- \u22a2 f x \u2208 u\n  exact hx\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f '' (f\u207b\u00b9' u) \u2286 u :=\nby\n  intros y h\n  -- y : \u03b2\n  -- h : y \u2208 f '' (f \u207b\u00b9' u)\n  -- \u22a2 y \u2208 u\n  rcases h with \u27e8x, hx, fxy\u27e9\n  -- x : \u03b1\n  -- hx : x \u2208 f \u207b\u00b9' u\n  -- fxy : f x = y\n  rw [\u2190fxy]\n  -- \u22a2 f x \u2208 u\n  exact hx\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f '' (f\u207b\u00b9' u) \u2286 u :=\nby\n  rintro y \u27e8x, hx, fxy\u27e9\n  -- y : \u03b2\n  -- x : \u03b1\n  -- hx : x \u2208 f \u207b\u00b9' u\n  -- fxy : f x = y\n  -- \u22a2 y \u2208 u\n  rw [\u2190fxy]\n  -- \u22a2 f x \u2208 u\n  exact hx\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f '' (f\u207b\u00b9' u) \u2286 u :=\nby\n  rintro y \u27e8x, hx, rfl\u27e9\n  -- x : \u03b1\n  -- hx : x \u2208 f \u207b\u00b9' u\n  -- \u22a2 f x \u2208 u\n  exact hx\n\n-- 6\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f '' (f\u207b\u00b9' u) \u2286 u :=\nimage_preimage_subset f u\n\n-- Lemas usados\n-- ============\n\n-- #check (image_preimage_subset f u : f '' (f\u207b\u00b9' u) \u2286 u)\n<\/pre>\n<p>Se puede interactuar con las demostraciones anteriores en <a href=\"https:\/\/live.lean-lang.org\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus2\/main\/src\/Imagen_de_la_imagen_inversa.lean\">Lean 4 Web<\/a>.<\/p>\n<h4>2.3. Demostraciones con Isabelle\/HOL<\/h4>\n<pre lang=\"isar\">\ntheory Imagen_de_la_imagen_inversa\nimports Main\nbegin\n\n(* 1\u00aa demostraci\u00f3n *)\n\nlemma \"f ` (f -` u) \u2286 u\"\nproof (rule subsetI)\n  fix y\n  assume \"y \u2208 f ` (f -` u)\"\n  then show \"y \u2208 u\"\n  proof (rule imageE)\n    fix x\n    assume \"y = f x\"\n    assume \"x \u2208 f -` u\"\n    then have \"f x \u2208 u\"\n      by (rule vimageD)\n    with \u2039y = f x\u203a show \"y \u2208 u\"\n      by (rule ssubst)\n  qed\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\n\nlemma \"f ` (f -` u) \u2286 u\"\nproof\n  fix y\n  assume \"y \u2208 f ` (f -` u)\"\n  then show \"y \u2208 u\"\n  proof\n    fix x\n    assume \"y = f x\"\n    assume \"x \u2208 f -` u\"\n    then have \"f x \u2208 u\"\n      by simp\n    with \u2039y = f x\u203a show \"y \u2208 u\"\n      by simp\n  qed\nqed\n\n(* 3\u00aa demostraci\u00f3n *)\n\nlemma \"f ` (f -` u) \u2286 u\"\n  by (simp only: image_vimage_subset)\n\n(* 4\u00aa demostraci\u00f3n *)\n\nlemma \"f ` (f -` u) \u2286 u\"\n  by auto\n\nend\n<\/pre>\n<p><a name=\"ej3\"><\/a><\/p>\n<h3>3. Si f es suprayectiva, entonces u \u2286 f[f\u207b\u00b9[u]]<\/h3>\n<p>Demostrar con Lean4 que si &#92;(f&#92;) es suprayectiva, entonces<br \/>\n&#92;[ u \u2286 f[f\u207b\u00b9[u]] &#92;]<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Function\nopen Set Function\nvariable {\u03b1 \u03b2 : Type _}\nvariable (f : \u03b1 \u2192 \u03b2)\nvariable (u : Set \u03b2)\n\nexample\n  (h : Surjective f)\n  : u \u2286 f '' (f\u207b\u00b9' u) :=\nby sorry\n<\/pre>\n<h4>3.1. Demostraci\u00f3n en lenguaje natural<\/h4>\n<p>Sea &#92;(y \u2208 u&#92;). Por ser &#92;(f&#92;) suprayectiva, exite un &#92;(x&#92;) tal que<br \/>\n&#92;[ f(x) = y &#92;tag{1} &#92;]<br \/>\nPor tanto, &#92;(x \u2208 f\u207b\u00b9[u]&#92;) y<br \/>\n&#92;[ f(x) \u2208 f[f\u207b\u00b9[u]] &#92;tag{2} &#92;]<br \/>\nFinalmente, por (1) y (2),<br \/>\n&#92;[ y \u2208 f[f\u207b\u00b9[u]] &#92;]<\/p>\n<h4>3.2. Demostraciones con Lean4<\/h4>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Function\nopen Set Function\nvariable {\u03b1 \u03b2 : Type _}\nvariable (f : \u03b1 \u2192 \u03b2)\nvariable (u : Set \u03b2)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : Surjective f)\n  : u \u2286 f '' (f\u207b\u00b9' u) :=\nby\n  intros y yu\n  -- y : \u03b2\n  -- yu : y \u2208 u\n  -- \u22a2 y \u2208 f '' (f \u207b\u00b9' u)\n  rcases h y with \u27e8x, fxy\u27e9\n  -- x : \u03b1\n  -- fxy : f x = y\n  use x\n  -- \u22a2 x \u2208 f \u207b\u00b9' u \u2227 f x = y\n  constructor\n  { -- \u22a2 x \u2208 f \u207b\u00b9' u\n    apply mem_preimage.mpr\n    -- \u22a2 f x \u2208 u\n    rw [fxy]\n    -- \u22a2 y \u2208 u\n    exact yu }\n  { -- \u22a2 f x = y\n    exact fxy }\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : Surjective f)\n  : u \u2286 f '' (f\u207b\u00b9' u) :=\nby\n  intros y yu\n  -- y : \u03b2\n  -- yu : y \u2208 u\n  -- \u22a2 y \u2208 f '' (f \u207b\u00b9' u)\n  rcases h y with \u27e8x, fxy\u27e9\n  -- x : \u03b1\n  -- fxy : f x = y\n  -- \u22a2 y \u2208 f '' (f \u207b\u00b9' u)\n  use x\n  -- \u22a2 x \u2208 f \u207b\u00b9' u \u2227 f x = y\n  constructor\n  { show f x \u2208 u\n    rw [fxy]\n    -- \u22a2 y \u2208 u\n    exact yu }\n  { show f x = y\n    exact fxy }\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : Surjective f)\n  : u \u2286 f '' (f\u207b\u00b9' u) :=\nby\n  intros y yu\n  -- y : \u03b2\n  -- yu : y \u2208 u\n  -- \u22a2 y \u2208 f '' (f \u207b\u00b9' u)\n  rcases h y with \u27e8x, fxy\u27e9\n  -- x : \u03b1\n  -- fxy : f x = y\n  aesop\n\n-- Lemas usados\n-- ============\n\n-- variable (x : \u03b1)\n-- #check (mem_preimage : x \u2208 f \u207b\u00b9' u \u2194 f x \u2208 u)\n<\/pre>\n<p>Se puede interactuar con las demostraciones anteriores en <a href=\"https:\/\/live.lean-lang.org\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus2\/main\/src\/Imagen_de_imagen_inversa_de_aplicaciones_suprayectivas.lean\" rel=\"noopener noreferrer\" target=\"_blank\">Lean 4 Web<\/a>.<\/p>\n<h4>3.3. Demostraciones con Isabelle\/HOL<\/h4>\n<pre lang=\"isar\">\ntheory Imagen_de_imagen_inversa_de_aplicaciones_suprayectivas\nimports Main\nbegin\n\n(* 1\u00aa demostraci\u00f3n *)\nlemma\n  assumes \"surj f\"\n  shows \"u \u2286 f ` (f -` u)\"\nproof (rule subsetI)\n  fix y\n  assume \"y \u2208 u\"\n  have \"\u2203x. y = f x\"\n    using \u2039surj f\u203a by (rule surjD)\n  then obtain x where \"y = f x\"\n    by (rule exE)\n  then have \"f x \u2208 u\"\n    using \u2039y \u2208 u\u203a by (rule subst)\n  then have \"x \u2208 f -` u\"\n    by (simp only: vimage_eq)\n  then have \"f x \u2208 f ` (f -` u)\"\n    by (rule imageI)\n  with \u2039y = f x\u203a show \"y \u2208 f ` (f -` u)\"\n    by (rule ssubst)\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\nlemma\n  assumes \"surj f\"\n  shows \"u \u2286 f ` (f -` u)\"\nproof\n  fix y\n  assume \"y \u2208 u\"\n  have \"\u2203x. y = f x\"\n    using \u2039surj f\u203a by (rule surjD)\n  then obtain x where \"y = f x\"\n    by (rule exE)\n  then have \"f x \u2208 u\"\n    using \u2039y \u2208 u\u203a by simp\n  then have \"x \u2208 f -` u\"\n    by simp\n  then have \"f x \u2208 f ` (f -` u)\"\n    by simp\n  with \u2039y = f x\u203a show \"y \u2208 f ` (f -` u)\"\n    by simp\nqed\n\n(* 3\u00aa demostraci\u00f3n *)\nlemma\n  assumes \"surj f\"\n  shows \"u \u2286 f ` (f -` u)\"\n  using assms\n  by (simp only: surj_image_vimage_eq)\n\n(* 4\u00aa demostraci\u00f3n *)\nlemma\n  assumes \"surj f\"\n  shows \"u \u2286 f ` (f -` u)\"\n  using assms\n  unfolding surj_def\n  by auto\n\n(* 5\u00aa demostraci\u00f3n *)\nlemma\n  assumes \"surj f\"\n  shows \"u \u2286 f ` (f -` u)\"\n  using assms\n  by auto\n\nend\n<\/pre>\n<p><a name=\"ej4\"><\/a><\/p>\n<h3>4. Si s \u2286 t, entonces f[s] \u2286 f[t]<\/h3>\n<p>Demostrar con Lean4 que si &#92;(s \u2286 t&#92;), entonces &#92;(f[s] \u2286 f[t]&#92;).<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Function\nimport Mathlib.Tactic\n\nopen Set\n\nvariable {\u03b1 \u03b2 : Type _}\nvariable (f : \u03b1 \u2192 \u03b2)\nvariable (s t : Set \u03b1)\n\nexample\n  (h : s \u2286 t)\n  : f '' s \u2286 f '' t :=\nby sorry\n<\/pre>\n<h4>4.1. Demostraci\u00f3n en lenguaje natural<\/h4>\n<p>Sea &#92;(y \u2208 f[s]&#92;). Entonces, existe un x tal que<br \/>\n&#92;begin{align}<br \/>\n   &amp;x \u2208 s    &#92;tag{1} &#92;&#92;<br \/>\n   &amp;f(x) = y &#92;tag{2}<br \/>\n&#92;end{align}<br \/>\nPor (1) y la hip\u00f3tesis,<br \/>\n&#92;[ x \u2208 t &#92;tag{3} &#92;]<br \/>\nPor (3),<br \/>\n&#92;[ f(x) \u2208 f[t] &#92;tag{4} &#92;]<br \/>\ny, por (2) y (4),<br \/>\n&#92;[ y \u2208 f[t] &#92;]<\/p>\n<h4>4.2. Demostraciones con Lean4<\/h4>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Function\nimport Mathlib.Tactic\n\nopen Set\n\nvariable {\u03b1 \u03b2 : Type _}\nvariable (f : \u03b1 \u2192 \u03b2)\nvariable (s t : Set \u03b1)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : s \u2286 t)\n  : f '' s \u2286 f '' t :=\nby\n  intros y hy\n  -- y : \u03b2\n  -- hy : y \u2208 f '' s\n  -- \u22a2 y \u2208 f '' t\n  rw [mem_image] at hy\n  -- hy : \u2203 x, x \u2208 s \u2227 f x = y\n  rcases hy with \u27e8x, hx\u27e9\n  -- x : \u03b1\n  -- hx : x \u2208 s \u2227 f x = y\n  rcases hx with \u27e8xs, fxy\u27e9\n  -- xs : x \u2208 s\n  -- fxy : f x = y\n  use x\n  -- \u22a2 x \u2208 t \u2227 f x = y\n  constructor\n  . -- \u22a2 x \u2208 t\n    exact h xs\n  . -- \u22a2 f x = y\n    exact fxy\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : s \u2286 t)\n  : f '' s \u2286 f '' t :=\nby\n  intros y hy\n  -- y : \u03b2\n  -- hy : y \u2208 f '' s\n  -- \u22a2 y \u2208 f '' t\n  rcases hy with \u27e8x, xs, fxy\u27e9\n  -- x : \u03b1\n  -- xs : x \u2208 s\n  -- fxy : f x = y\n  use x\n  -- \u22a2 x \u2208 t \u2227 f x = y\n  exact \u27e8h xs, fxy\u27e9\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : s \u2286 t)\n  : f '' s \u2286 f '' t :=\nimage_subset f h\n\n-- Lemas usados\n-- ============\n\n-- variable (y : \u03b2)\n-- #check (mem_image f s y : y \u2208 f '' s \u2194 \u2203 x, x \u2208 s \u2227 f x = y)\n-- #check (image_subset f : s \u2286 t \u2192 f '' s \u2286 f '' t)\n<\/pre>\n<p>Se puede interactuar con las demostraciones anteriores en <a href=\"https:\/\/live.lean-lang.org\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus2\/main\/src\/Monotonia_de_la_imagen_de_conjuntos.lean\" rel=\"noopener noreferrer\" target=\"_blank\">Lean 4 Web<\/a>.<\/p>\n<h4>4.3. Demostraciones con Isabelle\/HOL<\/h4>\n<pre lang=\"isar\">\ntheory Monotonia_de_la_imagen_de_conjuntos\nimports Main\nbegin\n\n(* 1\u00aa demostraci\u00f3n *)\nlemma\n  assumes \"s \u2286 t\"\n  shows \"f ` s \u2286 f ` t\"\nproof (rule subsetI)\n  fix y\n  assume \"y \u2208 f ` s\"\n  then show \"y \u2208 f ` t\"\n  proof (rule imageE)\n    fix x\n    assume \"y = f x\"\n    assume \"x \u2208 s\"\n    then have \"x \u2208 t\"\n      using \u2039s \u2286 t\u203a by (simp only: set_rev_mp)\n    then have \"f x \u2208 f ` t\"\n      by (rule imageI)\n    with \u2039y = f x\u203a show \"y \u2208 f ` t\"\n      by (rule ssubst)\n  qed\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\nlemma\n  assumes \"s \u2286 t\"\n  shows \"f ` s \u2286 f ` t\"\nproof\n  fix y\n  assume \"y \u2208 f ` s\"\n  then show \"y \u2208 f ` t\"\n  proof\n    fix x\n    assume \"y = f x\"\n    assume \"x \u2208 s\"\n    then have \"x \u2208 t\"\n      using \u2039s \u2286 t\u203a by (simp only: set_rev_mp)\n    then have \"f x \u2208 f ` t\"\n      by simp\n    with \u2039y = f x\u203a show \"y \u2208 f ` t\"\n      by simp\n  qed\nqed\n\n(* 3\u00aa demostraci\u00f3n *)\nlemma\n  assumes \"s \u2286 t\"\n  shows \"f ` s \u2286 f ` t\"\n  using assms\n  by blast\n\n(* 4\u00aa demostraci\u00f3n *)\nlemma\n  assumes \"s \u2286 t\"\n  shows \"f ` s \u2286 f ` t\"\n  using assms\n  by (simp only: image_mono)\n\nend\n<\/pre>\n<p><a name=\"ej5\"><\/a><\/p>\n<h3>5. Si u \u2286 v, entonces f\u207b\u00b9[u] \u2286 f\u207b\u00b9[v]<\/h3>\n<p>Demostrar con Lean4 que si &#92;(u \u2286 v&#92;), entonces &#92;(f\u207b\u00b9[u] \u2286 f\u207b\u00b9[v]&#92;).<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Function\nopen Set\n\nvariable {\u03b1 \u03b2 : Type _}\nvariable (f : \u03b1 \u2192 \u03b2)\nvariable (u v : Set \u03b2)\n\nexample\n  (h : u \u2286 v)\n  : f \u207b\u00b9' u \u2286 f \u207b\u00b9' v :=\nby sorry\n<\/pre>\n<h4>5.1. Demostraci\u00f3n en lenguaje natural<\/h4>\n<p>Por la siguiente cadena de implicaciones:<br \/>\n&#92;begin{align}<br \/>\n   x \u2208 f\u207b\u00b9[u] &amp;\u27f9 f(x) \u2208 u &#92;&#92;<br \/>\n              &amp;\u27f9 f(x) \u2208 v      &amp;&amp;&#92;text{[porque &#92;(u \u2286 v&#92;)]} &#92;&#92;<br \/>\n              &amp;\u27f9 x \u2208 f\u207b\u00b9[v]<br \/>\n&#92;end{align}<\/p>\n<h4>5.2. Demostraciones con Lean4<\/h4>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Function\nopen Set\n\nvariable {\u03b1 \u03b2 : Type _}\nvariable (f : \u03b1 \u2192 \u03b2)\nvariable (u v : Set \u03b2)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : u \u2286 v)\n  : f \u207b\u00b9' u \u2286 f \u207b\u00b9' v :=\nby\n  intros x hx\n  -- x : \u03b1\n  -- hx : x \u2208 f \u207b\u00b9' u\n  -- \u22a2 x \u2208 f \u207b\u00b9' v\n  have h1 : f x \u2208 u := mem_preimage.mp hx\n  have h2 : f x \u2208 v := h h1\n  show x \u2208 f \u207b\u00b9' v\n  exact mem_preimage.mpr h2\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : u \u2286 v)\n  : f \u207b\u00b9' u \u2286 f \u207b\u00b9' v :=\nby\n  intros x hx\n  -- x : \u03b1\n  -- hx : x \u2208 f \u207b\u00b9' u\n  -- \u22a2 x \u2208 f \u207b\u00b9' v\n  apply mem_preimage.mpr\n  -- \u22a2 f x \u2208 v\n  apply h\n  -- \u22a2 f x \u2208 u\n  apply mem_preimage.mp\n  -- \u22a2 x \u2208 f \u207b\u00b9' u\n  exact hx\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : u \u2286 v)\n  : f \u207b\u00b9' u \u2286 f \u207b\u00b9' v :=\nby\n  intros x hx\n  -- x : \u03b1\n  -- hx : x \u2208 f \u207b\u00b9' u\n  -- \u22a2 x \u2208 f \u207b\u00b9' v\n  apply h\n  -- \u22a2 f x \u2208 u\n  exact hx\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : u \u2286 v)\n  : f \u207b\u00b9' u \u2286 f \u207b\u00b9' v :=\nby\n  intros x hx\n  -- x : \u03b1\n  -- hx : x \u2208 f \u207b\u00b9' u\n  -- \u22a2 x \u2208 f \u207b\u00b9' v\n  exact h hx\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : u \u2286 v)\n  : f \u207b\u00b9' u \u2286 f \u207b\u00b9' v :=\nfun _ hx \u21a6 h hx\n\n-- 6\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : u \u2286 v)\n  : f \u207b\u00b9' u \u2286 f \u207b\u00b9' v :=\nby intro x; apply h\n\n-- 7\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : u \u2286 v)\n  : f \u207b\u00b9' u \u2286 f \u207b\u00b9' v :=\npreimage_mono h\n\n-- 8\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : u \u2286 v)\n  : f \u207b\u00b9' u \u2286 f \u207b\u00b9' v :=\nby tauto\n\n-- Lemas usados\n-- ============\n\n-- variable (a : \u03b1)\n-- #check (mem_preimage : a \u2208 f \u207b\u00b9' u \u2194 f a \u2208 u)\n-- #check (preimage_mono : u \u2286 v \u2192 f \u207b\u00b9' u \u2286 f \u207b\u00b9' v)\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\/Monotonia_de_la_imagen_inversa.lean\" rel=\"noopener noreferrer\" target=\"_blank\">Lean 4 Web<\/a>.<\/p>\n<h4>5.3. Demostraciones con Isabelle\/HOL<\/h4>\n<pre lang=\"isar\">\ntheory Monotonia_de_la_imagen_inversa\nimports Main\nbegin\n\n(* 1\u00aa demostraci\u00f3n *)\nlemma\n  assumes \"u \u2286 v\"\n  shows \"f -` u \u2286 f -` v\"\nproof (rule subsetI)\n  fix x\n  assume \"x \u2208 f -` u\"\n  then have \"f x \u2208 u\"\n    by (rule vimageD)\n  then have \"f x \u2208 v\"\n    using \u2039u \u2286 v\u203a by (rule set_rev_mp)\n  then show \"x \u2208 f -` v\"\n    by (simp only: vimage_eq)\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\nlemma\n  assumes \"u \u2286 v\"\n  shows \"f -` u \u2286 f -` v\"\nproof\n  fix x\n  assume \"x \u2208 f -` u\"\n  then have \"f x \u2208 u\"\n    by simp\n  then have \"f x \u2208 v\"\n    using \u2039u \u2286 v\u203a by (rule set_rev_mp)\n  then show \"x \u2208 f -` v\"\n    by simp\nqed\n\n(* 3\u00aa demostraci\u00f3n *)\nlemma\n  assumes \"u \u2286 v\"\n  shows \"f -` u \u2286 f -` v\"\n  using assms\n  by (simp only: vimage_mono)\n\n(* 4\u00aa demostraci\u00f3n *)\nlemma\n  assumes \"u \u2286 v\"\n  shows \"f -` u \u2286 f -` v\"\n  using assms\n  by blast\n\nend\n<\/pre>\n<p><a name=\"ej6\"><\/a><\/p>\n<h3>6. f\u207b\u00b9[A \u222a B] = f\u207b\u00b9[A] \u222a f\u207b\u00b9[B]<\/h3>\n<p>Demostrar con Lean4 que &#92;(f\u207b\u00b9[A \u222a B] = f\u207b\u00b9[A] \u222a f\u207b\u00b9[B]&#92;).<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Function\n\nopen Set\n\nvariable {\u03b1 \u03b2 : Type _}\nvariable (f : \u03b1 \u2192 \u03b2)\nvariable (A B : Set \u03b2)\n\nexample : f \u207b\u00b9' (A \u222a B) = f \u207b\u00b9' A \u222a f \u207b\u00b9' B :=\nby sorry\n<\/pre>\n<h4>6.1. Demostraci\u00f3n en lenguaje natural<\/h4>\n<p>Tenemos que demostrar que, para todo &#92;(x&#92;),<br \/>\n&#92;[ x \u2208 f\u207b\u00b9[A \u222a B] \u2194 x \u2208 f\u207b\u00b9[A] \u222a f\u207b\u00b9[B] &#92;]<br \/>\nLo haremos demostrando las dos implicaciones.<\/p>\n<p>(\u27f9) Supongamos que &#92;(x \u2208 f\u207b\u00b9[A \u222a B]&#92;). Entonces, &#92;(f(x) \u2208 A \u222a B&#92;).<br \/>\nDistinguimos dos casos:<\/p>\n<p>Caso 1: Supongamos que &#92;(f(x) \u2208 A&#92;). Entonces, &#92;(x \u2208 f\u207b\u00b9[A]&#92;) y, por tanto,<br \/>\n&#92;(x \u2208 f\u207b\u00b9[A] \u222a f\u207b\u00b9[B]&#92;).<\/p>\n<p>Caso 2: Supongamos que &#92;(f(x) \u2208 B&#92;). Entonces, &#92;(x \u2208 f\u207b\u00b9[B]&#92;) y, por tanto,<br \/>\n&#92;(x \u2208 f\u207b\u00b9[A] \u222a f\u207b\u00b9[B]&#92;).<\/p>\n<p>(\u27f8) Supongamos que &#92;(x \u2208 f\u207b\u00b9[A] \u222a f\u207b\u00b9[B]&#92;). Distinguimos dos casos.<\/p>\n<p>Caso 1: Supongamos que &#92;(x \u2208 f\u207b\u00b9[A]&#92;). Entonces, &#92;(f(x) \u2208 A&#92;) y, por tanto,<br \/>\n&#92;(f(x) \u2208 A \u222a B&#92;). Luego, &#92;(x \u2208 f\u207b\u00b9[A \u222a B]&#92;).<\/p>\n<p>Caso 2: Supongamos que &#92;(x \u2208 f\u207b\u00b9[B]&#92;). Entonces, &#92;(f(x) \u2208 B&#92;) y, por tanto,<br \/>\n&#92;(f(x) \u2208 A \u222a B&#92;). Luego, &#92;(x \u2208 f\u207b\u00b9[A \u222a B]&#92;).<\/p>\n<h4>6.2. Demostraciones con Lean4<\/h4>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Function\n\nopen Set\n\nvariable {\u03b1 \u03b2 : Type _}\nvariable (f : \u03b1 \u2192 \u03b2)\nvariable (A B : Set \u03b2)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f \u207b\u00b9' (A \u222a B) = f \u207b\u00b9' A \u222a f \u207b\u00b9' B :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 f \u207b\u00b9' (A \u222a B) \u2194 x \u2208 f \u207b\u00b9' A \u222a f \u207b\u00b9' B\n  constructor\n  . -- \u22a2 x \u2208 f \u207b\u00b9' (A \u222a B) \u2192 x \u2208 f \u207b\u00b9' A \u222a f \u207b\u00b9' B\n    intro h\n    -- h : x \u2208 f \u207b\u00b9' (A \u222a B)\n    -- \u22a2 x \u2208 f \u207b\u00b9' A \u222a f \u207b\u00b9' B\n    rw [mem_preimage] at h\n    -- h : f x \u2208 A \u222a B\n    rcases h with fxA | fxB\n    . -- fxA : f x \u2208 A\n      left\n      -- \u22a2 x \u2208 f \u207b\u00b9' A\n      apply mem_preimage.mpr\n      -- \u22a2 f x \u2208 A\n      exact fxA\n    . -- fxB : f x \u2208 B\n      right\n      -- \u22a2 x \u2208 f \u207b\u00b9' B\n      apply mem_preimage.mpr\n      -- \u22a2 f x \u2208 B\n      exact fxB\n  . -- \u22a2 x \u2208 f \u207b\u00b9' A \u222a f \u207b\u00b9' B \u2192 x \u2208 f \u207b\u00b9' (A \u222a B)\n    intro h\n    -- h : x \u2208 f \u207b\u00b9' A \u222a f \u207b\u00b9' B\n    -- \u22a2 x \u2208 f \u207b\u00b9' (A \u222a B)\n    rw [mem_preimage]\n    -- \u22a2 f x \u2208 A \u222a B\n    rcases h with xfA | xfB\n    . -- xfA : x \u2208 f \u207b\u00b9' A\n      rw [mem_preimage] at xfA\n      -- xfA : f x \u2208 A\n      left\n      -- \u22a2 f x \u2208 A\n      exact xfA\n    . -- xfB : x \u2208 f \u207b\u00b9' B\n      rw [mem_preimage] at xfB\n      -- xfB : f x \u2208 B\n      right\n      -- \u22a2 f x \u2208 B\n      exact xfB\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f \u207b\u00b9' (A \u222a B) = f \u207b\u00b9' A \u222a f \u207b\u00b9' B :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 f \u207b\u00b9' (A \u222a B) \u2194 x \u2208 f \u207b\u00b9' A \u222a f \u207b\u00b9' B\n  constructor\n  . -- \u22a2 x \u2208 f \u207b\u00b9' (A \u222a B) \u2192 x \u2208 f \u207b\u00b9' A \u222a f \u207b\u00b9' B\n    intros h\n    -- h : x \u2208 f \u207b\u00b9' (A \u222a B)\n    -- \u22a2 x \u2208 f \u207b\u00b9' A \u222a f \u207b\u00b9' B\n    rcases h with fxA | fxB\n    . -- fxA : f x \u2208 A\n      left\n      -- \u22a2 x \u2208 f \u207b\u00b9' A\n      exact fxA\n    . -- fxB : f x \u2208 B\n      right\n      -- \u22a2 x \u2208 f \u207b\u00b9' B\n      exact fxB\n  . -- \u22a2 x \u2208 f \u207b\u00b9' A \u222a f \u207b\u00b9' B \u2192 x \u2208 f \u207b\u00b9' (A \u222a B)\n    intro h\n    -- h : x \u2208 f \u207b\u00b9' A \u222a f \u207b\u00b9' B\n    -- \u22a2 x \u2208 f \u207b\u00b9' (A \u222a B)\n    rcases h with xfA | xfB\n    . -- xfA : x \u2208 f \u207b\u00b9' A\n      left\n      -- \u22a2 f x \u2208 A\n      exact xfA\n    . -- xfB : x \u2208 f \u207b\u00b9' B\n      right\n      -- \u22a2 f x \u2208 B\n      exact xfB\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f \u207b\u00b9' (A \u222a B) = f \u207b\u00b9' A \u222a f \u207b\u00b9' B :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 f \u207b\u00b9' (A \u222a B) \u2194 x \u2208 f \u207b\u00b9' A \u222a f \u207b\u00b9' B\n  constructor\n  . -- \u22a2 x \u2208 f \u207b\u00b9' (A \u222a B) \u2192 x \u2208 f \u207b\u00b9' A \u222a f \u207b\u00b9' B\n    rintro (fxA | fxB)\n    . -- fxA : f x \u2208 A\n      -- \u22a2 x \u2208 f \u207b\u00b9' A \u222a f \u207b\u00b9' B\n      exact Or.inl fxA\n    . -- fxB : f x \u2208 B\n      -- \u22a2 x \u2208 f \u207b\u00b9' A \u222a f \u207b\u00b9' B\n      exact Or.inr fxB\n  . -- \u22a2 x \u2208 f \u207b\u00b9' A \u222a f \u207b\u00b9' B \u2192 x \u2208 f \u207b\u00b9' (A \u222a B)\n    rintro (xfA | xfB)\n    . -- xfA : x \u2208 f \u207b\u00b9' A\n      -- \u22a2 x \u2208 f \u207b\u00b9' (A \u222a B)\n      exact Or.inl xfA\n    . -- xfB : x \u2208 f \u207b\u00b9' B\n      -- \u22a2 x \u2208 f \u207b\u00b9' (A \u222a B)\n      exact Or.inr xfB\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f \u207b\u00b9' (A \u222a B) = f \u207b\u00b9' A \u222a f \u207b\u00b9' B :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 f \u207b\u00b9' (A \u222a B) \u2194 x \u2208 f \u207b\u00b9' A \u222a f \u207b\u00b9' B\n  constructor\n  . -- \u22a2 x \u2208 f \u207b\u00b9' (A \u222a B) \u2192 x \u2208 f \u207b\u00b9' A \u222a f \u207b\u00b9' B\n    aesop\n  . -- \u22a2 x \u2208 f \u207b\u00b9' A \u222a f \u207b\u00b9' B \u2192 x \u2208 f \u207b\u00b9' (A \u222a B)\n    aesop\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f \u207b\u00b9' (A \u222a B) = f \u207b\u00b9' A \u222a f \u207b\u00b9' B :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 f \u207b\u00b9' (A \u222a B) \u2194 x \u2208 f \u207b\u00b9' A \u222a f \u207b\u00b9' B\n  aesop\n\n-- 6\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f \u207b\u00b9' (A \u222a B) = f \u207b\u00b9' A \u222a f \u207b\u00b9' B :=\nby ext ; aesop\n\n-- 7\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f \u207b\u00b9' (A \u222a B) = f \u207b\u00b9' A \u222a f \u207b\u00b9' B :=\nby ext ; rfl\n\n-- 8\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f \u207b\u00b9' (A \u222a B) = f \u207b\u00b9' A \u222a f \u207b\u00b9' B :=\nrfl\n\n-- 9\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f \u207b\u00b9' (A \u222a B) = f \u207b\u00b9' A \u222a f \u207b\u00b9' B :=\npreimage_union\n\n-- 10\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f \u207b\u00b9' (A \u222a B) = f \u207b\u00b9' A \u222a f \u207b\u00b9' B :=\nby simp\n\n-- Lemas usados\n-- ============\n\n-- variable (x : \u03b1)\n-- variable (p q : Prop)\n-- #check (Or.inl: p \u2192 p \u2228 q)\n-- #check (Or.inr: q \u2192 p \u2228 q)\n-- #check (mem_preimage : x \u2208 f \u207b\u00b9' A \u2194 f x \u2208 A)\n-- #check (preimage_union : f \u207b\u00b9' (A \u222a B) = f \u207b\u00b9' A \u222a f \u207b\u00b9' 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\/Imagen_inversa_de_la_union.lean\" rel=\"noopener noreferrer\" target=\"_blank\">Lean 4 Web<\/a>.<\/p>\n<h4>6.3. Demostraciones con Isabelle\/HOL<\/h4>\n<pre lang=\"isar\">\ntheory Imagen_inversa_de_la_union\nimports Main\nbegin\n\n(* 1\u00aa demostraci\u00f3n *)\n\nlemma \"f -` (u \u222a v) = f -` u \u222a f -` v\"\nproof (rule equalityI)\n  show \"f -` (u \u222a v) \u2286 f -` u \u222a f -` v\"\n  proof (rule subsetI)\n    fix x\n    assume \"x \u2208 f -` (u \u222a v)\"\n    then have \"f x \u2208 u \u222a v\"\n      by (rule vimageD)\n    then show \"x \u2208 f -` u \u222a f -` v\"\n    proof (rule UnE)\n      assume \"f x \u2208 u\"\n      then have \"x \u2208 f -` u\"\n        by (rule vimageI2)\n      then show \"x \u2208 f -` u \u222a f -` v\"\n        by (rule UnI1)\n    next\n      assume \"f x \u2208 v\"\n      then have \"x \u2208 f -` v\"\n        by (rule vimageI2)\n      then show \"x \u2208 f -` u \u222a f -` v\"\n        by (rule UnI2)\n    qed\n  qed\nnext\n  show \"f -` u \u222a f -` v \u2286 f -` (u \u222a v)\"\n  proof (rule subsetI)\n    fix x\n    assume \"x \u2208 f -` u \u222a f -` v\"\n    then show \"x \u2208 f -` (u \u222a v)\"\n    proof (rule UnE)\n      assume \"x \u2208 f -` u\"\n      then have \"f x \u2208 u\"\n        by (rule vimageD)\n      then have \"f x \u2208 u \u222a v\"\n        by (rule UnI1)\n      then show \"x \u2208 f -` (u \u222a v)\"\n        by (rule vimageI2)\n    next\n      assume \"x \u2208 f -` v\"\n      then have \"f x \u2208 v\"\n        by (rule vimageD)\n      then have \"f x \u2208 u \u222a v\"\n        by (rule UnI2)\n      then show \"x \u2208 f -` (u \u222a v)\"\n        by (rule vimageI2)\n    qed\n  qed\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\n\nlemma \"f -` (u \u222a v) = f -` u \u222a f -` v\"\nproof\n  show \"f -` (u \u222a v) \u2286 f -` u \u222a f -` v\"\n  proof\n    fix x\n    assume \"x \u2208 f -` (u \u222a v)\"\n    then have \"f x \u2208 u \u222a v\" by simp\n    then show \"x \u2208 f -` u \u222a f -` v\"\n    proof\n      assume \"f x \u2208 u\"\n      then have \"x \u2208 f -` u\" by simp\n      then show \"x \u2208 f -` u \u222a f -` v\" by simp\n    next\n      assume \"f x \u2208 v\"\n      then have \"x \u2208 f -` v\" by simp\n      then show \"x \u2208 f -` u \u222a f -` v\" by simp\n    qed\n  qed\nnext\n  show \"f -` u \u222a f -` v \u2286 f -` (u \u222a v)\"\n  proof\n    fix x\n    assume \"x \u2208 f -` u \u222a f -` v\"\n    then show \"x \u2208 f -` (u \u222a v)\"\n    proof\n      assume \"x \u2208 f -` u\"\n      then have \"f x \u2208 u\" by simp\n      then have \"f x \u2208 u \u222a v\" by simp\n      then show \"x \u2208 f -` (u \u222a v)\" by simp\n    next\n      assume \"x \u2208 f -` v\"\n      then have \"f x \u2208 v\" by simp\n      then have \"f x \u2208 u \u222a v\" by simp\n      then show \"x \u2208 f -` (u \u222a v)\" by simp\n    qed\n  qed\nqed\n\n(* 3\u00aa demostraci\u00f3n *)\n\nlemma \"f -` (u \u222a v) = f -` u \u222a f -` v\"\n  by (simp only: vimage_Un)\n\n(* 4\u00aa demostraci\u00f3n *)\n\nlemma \"f -` (u \u222a v) = f -` u \u222a f -` v\"\n  by auto\n\nend\n<\/pre>\n<p><a name=\"ej7\"><\/a><\/p>\n<h3>7. f[s \u2229 t] \u2286 f[s] \u2229 f[t]<\/h3>\n<p>Demostrar con Lean4 que<br \/>\n&#92;[ f[s \u2229 t] \u2286 f[s] \u2229 f[t]\u200b &#92;]<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Function\nimport Mathlib.Tactic\n\nopen Set\n\nvariable {\u03b1 \u03b2 : Type _}\nvariable (f : \u03b1 \u2192 \u03b2)\nvariable (s t : Set \u03b1)\n\nexample : f '' (s \u2229 t) \u2286 f '' s \u2229 f '' t :=\nby sorry\n<\/pre>\n<h4>7.1. Demostraci\u00f3n en lenguaje natural<\/h4>\n<p>Sea tal que<br \/>\n&#92;[ y \u2208 f[s \u2229 t] &#92;]<br \/>\nPor tanto, existe un &#92;(x&#92;) tal que<br \/>\n&#92;begin{align}<br \/>\n  x \u2208 s \u2229 t  &#92;tag{1} &#92;&#92;<br \/>\n  f(x) = y   &#92;tag{2}<br \/>\n&#92;end{align}<br \/>\nPor (1), se tiene que<br \/>\n&#92;begin{align}<br \/>\n  x \u2208 s      &#92;tag{3} &#92;&#92;<br \/>\n  x \u2208 t      &#92;tag{4}<br \/>\n&#92;end{align}<br \/>\nPor (2) y (3), se tiene<br \/>\n&#92;[ y \u2208 f[s] &#92;tag{5} &#92;]<br \/>\nPor (2) y (4), se tiene<br \/>\n&#92;[ y \u2208 f[t] &#92;tag{6} &#92;]<br \/>\nPor (5) y (6), se tiene<br \/>\n&#92;[ y \u2208 f[s] \u2229 f[t] &#92;]<\/p>\n<h4>7.2. Demostraciones con Lean4<\/h4>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Function\nimport Mathlib.Tactic\n\nopen Set\n\nvariable {\u03b1 \u03b2 : Type _}\nvariable (f : \u03b1 \u2192 \u03b2)\nvariable (s t : Set \u03b1)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f '' (s \u2229 t) \u2286 f '' s \u2229 f '' t :=\nby\n  intros y hy\n  -- y : \u03b2\n  -- hy : y \u2208 f '' (s \u2229 t)\n  -- \u22a2 y \u2208 f '' s \u2229 f '' t\n  rcases hy with \u27e8x, hx\u27e9\n  -- x : \u03b1\n  -- hx : x \u2208 s \u2229 t \u2227 f x = y\n  rcases hx with \u27e8xst, fxy\u27e9\n  -- xst : x \u2208 s \u2229 t\n  -- fxy : f x = y\n  constructor\n  . -- \u22a2 y \u2208 f '' s\n    use x\n    -- \u22a2 x \u2208 s \u2227 f x = y\n    constructor\n    . -- \u22a2 x \u2208 s\n      exact xst.1\n    . -- \u22a2 f x = y\n      exact fxy\n  . -- \u22a2 y \u2208 f '' t\n    use x\n    -- \u22a2 x \u2208 t \u2227 f x = y\n    constructor\n    . -- \u22a2 x \u2208 t\n      exact xst.2\n    . -- \u22a2 f x = y\n      exact fxy\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f '' (s \u2229 t) \u2286 f '' s \u2229 f '' t :=\nby\n  intros y hy\n  -- y : \u03b2\n  -- hy : y \u2208 f '' (s \u2229 t)\n  -- \u22a2 y \u2208 f '' s \u2229 f '' t\n  rcases hy with \u27e8x, \u27e8xs, xt\u27e9, fxy\u27e9\n  -- x : \u03b1\n  -- fxy : f x = y\n  -- xs : x \u2208 s\n  -- xt : x \u2208 t\n  constructor\n  . -- \u22a2 y \u2208 f '' s\n    use x\n    -- \u22a2 x \u2208 s \u2227 f x = y\n    exact \u27e8xs, fxy\u27e9\n  . -- \u22a2 y \u2208 f '' t\n    use x\n    -- \u22a2 x \u2208 t \u2227 f x = y\n    exact \u27e8xt, fxy\u27e9\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f '' (s \u2229 t) \u2286 f '' s \u2229 f '' t :=\nimage_inter_subset f s t\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f '' (s \u2229 t) \u2286 f '' s \u2229 f '' t :=\nby intro ; aesop\n\n-- Lemas usados\n-- ============\n\n-- #check (image_inter_subset f s t : f '' (s \u2229 t) \u2286 f '' s \u2229 f '' t)\n<\/pre>\n<p>Se puede interactuar con las demostraciones anteriores en <a href=\"https:\/\/live.lean-lang.org\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus2\/main\/src\/Imagen_de_la_interseccion.lean\" rel=\"noopener noreferrer\" target=\"_blank\">Lean 4 Web<\/a>.<\/p>\n<h4>6.3. Demostraciones con Isabelle\/HOL<\/h4>\n<pre lang=\"isar\">\ntheory Imagen_de_la_interseccion\nimports Main\nbegin\n\n(* 1\u00aa demostraci\u00f3n *)\n\nlemma \"f ` (s \u2229 t) \u2286 f ` s \u2229 f ` t\"\nproof (rule subsetI)\n  fix y\n  assume \"y \u2208 f ` (s \u2229 t)\"\n  then have \"y \u2208 f ` s\"\n  proof (rule imageE)\n    fix x\n    assume \"y = f x\"\n    assume \"x \u2208 s \u2229 t\"\n    have \"x \u2208 s\"\n      using \u2039x \u2208 s \u2229 t\u203a by (rule IntD1)\n    then have \"f x \u2208 f ` s\"\n      by (rule imageI)\n    with \u2039y = f x\u203a show \"y \u2208 f ` s\"\n      by (rule ssubst)\n  qed\n  moreover\n  note \u2039y \u2208 f ` (s \u2229 t)\u203a\n  then have \"y \u2208 f ` t\"\n  proof (rule imageE)\n    fix x\n    assume \"y = f x\"\n    assume \"x \u2208 s \u2229 t\"\n    have \"x \u2208 t\"\n      using \u2039x \u2208 s \u2229 t\u203a by (rule IntD2)\n    then have \"f x \u2208 f ` t\"\n      by (rule imageI)\n    with \u2039y = f x\u203a show \"y \u2208 f ` t\"\n      by (rule ssubst)\n  qed\n  ultimately show \"y \u2208 f ` s \u2229 f ` t\"\n    by (rule IntI)\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\n\nlemma \"f ` (s \u2229 t) \u2286 f ` s \u2229 f ` t\"\nproof\n  fix y\n  assume \"y \u2208 f ` (s \u2229 t)\"\n  then have \"y \u2208 f ` s\"\n  proof\n    fix x\n    assume \"y = f x\"\n    assume \"x \u2208 s \u2229 t\"\n    have \"x \u2208 s\"\n      using \u2039x \u2208 s \u2229 t\u203a by simp\n    then have \"f x \u2208 f ` s\"\n      by simp\n    with \u2039y = f x\u203a show \"y \u2208 f ` s\"\n      by simp\n  qed\n  moreover\n  note \u2039y \u2208 f ` (s \u2229 t)\u203a\n  then have \"y \u2208 f ` t\"\n  proof\n    fix x\n    assume \"y = f x\"\n    assume \"x \u2208 s \u2229 t\"\n    have \"x \u2208 t\"\n      using \u2039x \u2208 s \u2229 t\u203a by simp\n    then have \"f x \u2208 f ` t\"\n      by simp\n    with \u2039y = f x\u203a show \"y \u2208 f ` t\"\n      by simp\n  qed\n  ultimately show \"y \u2208 f ` s \u2229 f ` t\"\n    by simp\nqed\n\n(* 3\u00aa demostraci\u00f3n *)\n\nlemma \"f ` (s \u2229 t) \u2286 f ` s \u2229 f ` t\"\nproof\n  fix y\n  assume \"y \u2208 f ` (s \u2229 t)\"\n  then obtain x where hx : \"y = f x \u2227 x \u2208 s \u2229 t\" by auto\n  then have \"y = f x\" by simp\n  have \"x \u2208 s\" using hx by simp\n  have \"x \u2208 t\" using hx by simp\n  have \"y \u2208  f ` s\" using \u2039y = f x\u203a \u2039x \u2208 s\u203a by simp\n  moreover\n  have \"y \u2208  f ` t\" using \u2039y = f x\u203a \u2039x \u2208 t\u203a by simp\n  ultimately show \"y \u2208 f ` s \u2229 f ` t\"\n    by simp\nqed\n\n(* 4\u00aa demostraci\u00f3n *)\n\nlemma \"f ` (s \u2229 t) \u2286 f ` s \u2229 f ` t\"\n  by (simp only: image_Int_subset)\n\n(* 5\u00aa demostraci\u00f3n *)\n\nlemma \"f ` (s \u2229 t) \u2286 f ` s \u2229 f ` t\"\n  by auto\n\nend\n<\/pre>\n<p><a name=\"ej8\"><\/a><\/p>\n<h3>8. Si f es inyectiva, entonces f[s] \u2229 f[t] \u2286 f[s \u2229 t]<\/h3>\n<p>Demostrar con Lean4 que si &#92;(f&#92;) es inyectiva, entonces<br \/>\n&#92;[ f[s] \u2229 f[t] \u2286 f[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.Function\n\nopen Set Function\n\nvariable {\u03b1 \u03b2 : Type _}\nvariable (f : \u03b1 \u2192 \u03b2)\nvariable (s t : Set \u03b1)\n\nexample\n  (h : Injective f)\n  : f '' s \u2229 f '' t \u2286 f '' (s \u2229 t) :=\nby sorry\n<\/pre>\n<h4>8.1. Demostraci\u00f3n en lenguaje natural<\/h4>\n<p>Sea &#92;(y \u2208 f[s] \u2229 f[t]&#92;). Entonces, existen &#92;(x\u2081&#92;) y &#92;(x\u2082&#92;) tales que<br \/>\n&#92;begin{align}<br \/>\n   &amp;x\u2081 \u2208 s      &#92;tag{1} &#92;&#92;<br \/>\n   &amp;f(x\u2081) = y   &#92;tag{2} &#92;&#92;<br \/>\n   &amp;x\u2082 \u2208 t      &#92;tag{3} &#92;&#92;<br \/>\n   &amp;f(x\u2082) = y   &#92;tag{4}<br \/>\n&#92;end{align}<br \/>\nDe (2) y (4) se tiene que<br \/>\n&#92;[ f(x\u2081) = f(x\u2082) &#92;]<br \/>\ny, por ser &#92;(f&#92;) inyectiva, se tiene que<br \/>\n&#92;[ x\u2081 = x\u2082 &#92;]<br \/>\ny, por (1), se tiene que<br \/>\n&#92;[ x\u2082 \u2208 t &#92;]<br \/>\ny, por (3), se tiene que<br \/>\n&#92;[ x\u2082 \u2208 s \u2229 t &#92;]<br \/>\nPor tanto,<br \/>\n&#92;[ f(x\u2082) \u2208 f[s \u2229 t] &#92;]<br \/>\ny, por (4),<br \/>\n&#92;[ y \u2208 f[s \u2229 t] &#92;]<\/p>\n<h4>8.2. Demostraciones con Lean4<\/h4>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Function\n\nopen Set Function\n\nvariable {\u03b1 \u03b2 : Type _}\nvariable (f : \u03b1 \u2192 \u03b2)\nvariable (s t : Set \u03b1)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : Injective f)\n  : f '' s \u2229 f '' t \u2286 f '' (s \u2229 t) :=\nby\n  intros y hy\n  -- y : \u03b2\n  -- hy : y \u2208 f '' s \u2229 f '' t\n  -- \u22a2 y \u2208 f '' (s \u2229 t)\n  rcases hy with \u27e8hy1, hy2\u27e9\n  -- hy1 : y \u2208 f '' s\n  -- hy2 : y \u2208 f '' t\n  rcases hy1 with \u27e8x1, hx1\u27e9\n  -- x1 : \u03b1\n  -- hx1 : x1 \u2208 s \u2227 f x1 = y\n  rcases hx1 with \u27e8x1s, fx1y\u27e9\n  -- x1s : x1 \u2208 s\n  -- fx1y : f x1 = y\n  rcases hy2 with \u27e8x2, hx2\u27e9\n  -- x2 : \u03b1\n  -- hx2 : x2 \u2208 t \u2227 f x2 = y\n  rcases hx2 with \u27e8x2t, fx2y\u27e9\n  -- x2t : x2 \u2208 t\n  -- fx2y : f x2 = y\n  have h1 : f x1 = f x2 := Eq.trans fx1y fx2y.symm\n  have h2 : x1 = x2 := h (congrArg f (h h1))\n  have h3 : x2 \u2208 s := by rwa [h2] at x1s\n  have h4 : x2 \u2208 s \u2229 t := by exact \u27e8h3, x2t\u27e9\n  have h5 : f x2 \u2208 f '' (s \u2229 t) := mem_image_of_mem f h4\n  show y \u2208 f '' (s \u2229 t)\n  rwa [fx2y] at h5\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : Injective f)\n  : f '' s \u2229 f '' t \u2286 f '' (s \u2229 t) :=\nby\n  intros y hy\n  -- y : \u03b2\n  -- hy : y \u2208 f '' s \u2229 f '' t\n  -- \u22a2 y \u2208 f '' (s \u2229 t)\n  rcases hy  with \u27e8hy1, hy2\u27e9\n  -- hy1 : y \u2208 f '' s\n  -- hy2 : y \u2208 f '' t\n  rcases hy1 with \u27e8x1, hx1\u27e9\n  -- x1 : \u03b1\n  -- hx1 : x1 \u2208 s \u2227 f x1 = y\n  rcases hx1 with \u27e8x1s, fx1y\u27e9\n  -- x1s : x1 \u2208 s\n  -- fx1y : f x1 = y\n  rcases hy2 with \u27e8x2, hx2\u27e9\n  -- x2 : \u03b1\n  -- hx2 : x2 \u2208 t \u2227 f x2 = y\n  rcases hx2 with \u27e8x2t, fx2y\u27e9\n  -- x2t : x2 \u2208 t\n  -- fx2y : f x2 = y\n  use x1\n  -- \u22a2 x1 \u2208 s \u2229 t \u2227 f x1 = y\n  constructor\n  . -- \u22a2 x1 \u2208 s \u2229 t\n    constructor\n    . -- \u22a2 x1 \u2208 s\n      exact x1s\n    . -- \u22a2 x1 \u2208 t\n      convert x2t\n      -- \u22a2 x1 = x2\n      apply h\n      -- \u22a2 f x1 = f x2\n      rw [\u2190 fx2y] at fx1y\n      -- fx1y : f x1 = f x2\n      exact fx1y\n  . -- \u22a2 f x1 = y\n    exact fx1y\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : Injective f)\n  : f '' s \u2229 f '' t \u2286 f '' (s \u2229 t) :=\nby\n  rintro y \u27e8\u27e8x1, x1s, fx1y\u27e9, \u27e8x2, x2t, fx2y\u27e9\u27e9\n  -- y : \u03b2\n  -- x1 : \u03b1\n  -- x1s : x1 \u2208 s\n  -- fx1y : f x1 = y\n  -- x2 : \u03b1\n  -- x2t : x2 \u2208 t\n  -- fx2y : f x2 = y\n  -- \u22a2 y \u2208 f '' (s \u2229 t)\n  use x1\n  -- \u22a2 x1 \u2208 s \u2229 t \u2227 f x1 = y\n  constructor\n  . -- \u22a2 x1 \u2208 s \u2229 t\n    constructor\n    . -- \u22a2 x1 \u2208 s\n      exact x1s\n    . -- \u22a2 x1 \u2208 t\n      convert x2t\n      -- \u22a2 x1 = x2\n      apply h\n      -- \u22a2 f x1 = f x2\n      rw [\u2190 fx2y] at fx1y\n      -- fx1y : f x1 = f x2\n      exact fx1y\n  . -- \u22a2 f x1 = y\n    exact fx1y\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\/Imagen_de_la_interseccion_de_aplicaciones_inyectivas.lean\" rel=\"noopener noreferrer\" target=\"_blank\">Lean 4 Web<\/a>.<\/p>\n<h4>8.3. Demostraciones con Isabelle\/HOL<\/h4>\n<pre lang=\"isar\">\ntheory Imagen_de_la_interseccion_de_aplicaciones_inyectivas\nimports Main\nbegin\n\n(* 1\u00aa demostraci\u00f3n *)\n\nlemma\n  assumes \"inj f\"\n  shows \"f ` s \u2229 f ` t \u2286 f ` (s \u2229 t)\"\nproof (rule subsetI)\n  fix y\n  assume \"y \u2208 f ` s \u2229 f ` t\"\n  then have \"y \u2208 f ` s\"\n    by (rule IntD1)\n  then show \"y \u2208 f ` (s \u2229 t)\"\n  proof (rule imageE)\n    fix x\n    assume \"y = f x\"\n    assume \"x \u2208 s\"\n    have \"x \u2208 t\"\n    proof -\n      have \"y \u2208 f ` t\"\n        using \u2039y \u2208 f ` s \u2229 f ` t\u203a by (rule IntD2)\n      then show \"x \u2208 t\"\n      proof (rule imageE)\n        fix z\n        assume \"y = f z\"\n        assume \"z \u2208 t\"\n        have \"f x = f z\"\n          using \u2039y = f x\u203a \u2039y = f z\u203a by (rule subst)\n        with \u2039inj f\u203a have \"x = z\"\n          by (simp only: inj_eq)\n        then show \"x \u2208 t\"\n          using \u2039z \u2208 t\u203a by (rule ssubst)\n      qed\n    qed\n    with \u2039x \u2208 s\u203a have \"x \u2208 s \u2229 t\"\n      by (rule IntI)\n    with \u2039y = f x\u203a show \"y \u2208 f ` (s \u2229 t)\"\n      by (rule image_eqI)\n  qed\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\n\nlemma\n  assumes \"inj f\"\n  shows \"f ` s \u2229 f ` t \u2286 f ` (s \u2229 t)\"\nproof\n  fix y\n  assume \"y \u2208 f ` s \u2229 f ` t\"\n  then have \"y \u2208 f ` s\" by simp\n  then show \"y \u2208 f ` (s \u2229 t)\"\n  proof\n    fix x\n    assume \"y = f x\"\n    assume \"x \u2208 s\"\n    have \"x \u2208 t\"\n    proof -\n      have \"y \u2208 f ` t\" using \u2039y \u2208 f ` s \u2229 f ` t\u203a by simp\n      then show \"x \u2208 t\"\n      proof\n        fix z\n        assume \"y = f z\"\n        assume \"z \u2208 t\"\n        have \"f x = f z\" using \u2039y = f x\u203a \u2039y = f z\u203a by simp\n        with \u2039inj f\u203a have \"x = z\" by (simp only: inj_eq)\n        then show \"x \u2208 t\" using \u2039z \u2208 t\u203a by simp\n      qed\n    qed\n    with \u2039x \u2208 s\u203a have \"x \u2208 s \u2229 t\" by simp\n    with \u2039y = f x\u203a show \"y \u2208 f ` (s \u2229 t)\" by simp\n  qed\nqed\n\n(* 3\u00aa demostraci\u00f3n *)\n\nlemma\n  assumes \"inj f\"\n  shows \"f ` s \u2229 f ` t \u2286 f ` (s \u2229 t)\"\n  using assms\n  by (simp only: image_Int)\n\n(* 4\u00aa demostraci\u00f3n *)\n\nlemma\n  assumes \"inj f\"\n  shows \"f ` s \u2229 f ` t \u2286 f ` (s \u2229 t)\"\n  using assms\n  unfolding inj_def\n  by auto\n\nend\n<\/pre>\n<p><a name=\"ej9\"><\/a><\/p>\n<h3>9. f[s] \\ f[t] \u2286 f[s \\ t]<\/h3>\n<p>Demostrar con Lean4 que<br \/>\n&#92;[f[s] &#92;setminus f[t] \u2286 f[s &#92;setminus t] &#92;]<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Function\nimport Mathlib.Tactic\n\nopen Set\n\nvariable {\u03b1 \u03b2 : Type _}\nvariable (f : \u03b1 \u2192 \u03b2)\nvariable (s t : Set \u03b1)\n\nexample : f '' s \\ f '' t \u2286 f '' (s \\ t) :=\nby sorry\n<\/pre>\n<h4>9.1. Demostraci\u00f3n en lenguaje natural<\/h4>\n<p>Sea &#92;(y \u2208 f[s] &#92;setminus f[t]&#92;). Entonces,<br \/>\n&#92;begin{align}<br \/>\n   &amp;y \u2208 f[s] &#92;tag{1} &#92;&#92;<br \/>\n   &amp;y \u2209 f[t] &#92;tag{2}<br \/>\n&#92;end{align}<br \/>\nPor (1), existe un &#92;(x&#92;) tal que<br \/>\n&#92;begin{align}<br \/>\n   &amp;x \u2208 s    &#92;tag{3} &#92;&#92;<br \/>\n   &amp;f(x) = y &#92;tag{4}<br \/>\n&#92;end{align}<br \/>\nPor tanto, para demostrar que &#92;(y \u2208 f[s &#92;setminus t]&#92;), basta probar que &#92;(x \u2209 t&#92;). Para ello, supongamos que &#92;(x \u2208 t&#92;). Entonces, por (4), &#92;(y \u2208 f[t]&#92;), en contradicci\u00f3n con (2).<\/p>\n<h4>9.2. Demostraciones con Lean4<\/h4>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Function\nimport Mathlib.Tactic\n\nopen Set\n\nvariable {\u03b1 \u03b2 : Type _}\nvariable (f : \u03b1 \u2192 \u03b2)\nvariable (s t : Set \u03b1)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f '' s \\ f '' t \u2286 f '' (s \\ t) :=\nby\n  intros y hy\n  -- y : \u03b2\n  -- hy : y \u2208 f '' s \\ f '' t\n  -- \u22a2 y \u2208 f '' (s \\ t)\n  rcases hy with \u27e8yfs, ynft\u27e9\n  -- yfs : y \u2208 f '' s\n  -- ynft : \u00acy \u2208 f '' t\n  rcases yfs with \u27e8x, hx\u27e9\n  -- x : \u03b1\n  -- hx : x \u2208 s \u2227 f x = y\n  rcases hx with \u27e8xs, fxy\u27e9\n  -- xs : x \u2208 s\n  -- fxy : f x = y\n  have h1 : x \u2209 t := by\n    intro xt\n    -- xt : x \u2208 t\n    -- \u22a2 False\n    have h2 : f x \u2208 f '' t := mem_image_of_mem f xt\n    have h3 : y \u2208 f '' t := by rwa [fxy] at h2\n    show False\n    exact ynft h3\n  have h4 : x \u2208 s \\ t := mem_diff_of_mem xs h1\n  have h5 : f x \u2208 f '' (s \\ t) := mem_image_of_mem f h4\n  show y \u2208 f '' (s \\ t)\n  rwa [fxy] at h5\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f '' s \\ f '' t \u2286 f '' (s \\ t) :=\nby\n  intros y hy\n  -- y : \u03b2\n  -- hy : y \u2208 f '' s \\ f '' t\n  -- \u22a2 y \u2208 f '' (s \\ t)\n  rcases hy with \u27e8yfs, ynft\u27e9\n  -- yfs : y \u2208 f '' s\n  -- ynft : \u00acy \u2208 f '' t\n  rcases yfs with \u27e8x, hx\u27e9\n  -- x : \u03b1\n  -- hx : x \u2208 s \u2227 f x = y\n  rcases hx with \u27e8xs, fxy\u27e9\n  -- xs : x \u2208 s\n  -- fxy : f x = y\n  use x\n  -- \u22a2 x \u2208 s \\ t \u2227 f x = y\n  constructor\n  . -- \u22a2 x \u2208 s \\ t\n    constructor\n    . -- \u22a2 x \u2208 s\n      exact xs\n    . -- \u22a2 \u00acx \u2208 t\n      intro xt\n      -- xt : x \u2208 t\n      -- \u22a2 False\n      apply ynft\n      -- \u22a2 y \u2208 f '' t\n      rw [\u2190fxy]\n      -- \u22a2 f x \u2208 f '' t\n      apply mem_image_of_mem\n      -- \u22a2 x \u2208 t\n      exact xt\n  . -- \u22a2 f x = y\n    exact fxy\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f '' s \\ f '' t \u2286 f '' (s \\ t) :=\nby\n  rintro y \u27e8\u27e8x, xs, fxy\u27e9, ynft\u27e9\n  -- y : \u03b2\n  -- ynft : \u00acy \u2208 f '' t\n  -- x : \u03b1\n  -- xs : x \u2208 s\n  -- fxy : f x = y\n  -- \u22a2 y \u2208 f '' (s \\ t)\n  use x\n  -- \u22a2 x \u2208 s \\ t \u2227 f x = y\n  aesop\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f '' s \\ f '' t \u2286 f '' (s \\ t) :=\nfun y \u27e8\u27e8x, xs, fxy\u27e9, ynft\u27e9 \u21a6 \u27e8x, by aesop\u27e9\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f '' s \\ f '' t \u2286 f '' (s \\ t) :=\nsubset_image_diff f s t\n\n-- Lemmas usados\n-- =============\n\n-- variable (x : \u03b1)\n-- #check (mem_image_of_mem f : x  \u2208 s \u2192 f x \u2208 f '' s)\n-- #check (subset_image_diff f s t : f '' s \\ f '' t \u2286 f '' (s \\ t))\n<\/pre>\n<p>Se puede interactuar con las demostraciones anteriores en <a href=\"https:\/\/live.lean-lang.org\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus2\/main\/src\/Imagen_de_la_diferencia_de_conjuntos.lean\" rel=\"noopener noreferrer\" target=\"_blank\">Lean 4 Web<\/a>.<\/p>\n<h4>9.3. Demostraciones con Isabelle\/HOL<\/h4>\n<pre lang=\"isar\">\ntheory Imagen_de_la_diferencia_de_conjuntos\nimports Main\nbegin\n\n(* 1\u00aa demostraci\u00f3n *)\n\nlemma \"f ` s - f ` t \u2286 f ` (s - t)\"\nproof (rule subsetI)\n  fix y\n  assume hy : \"y \u2208 f ` s - f ` t\"\n  then show \"y \u2208 f ` (s - t)\"\n  proof (rule DiffE)\n    assume \"y \u2208 f ` s\"\n    assume \"y \u2209 f ` t\"\n    note \u2039y \u2208 f ` s\u203a\n    then show \"y \u2208 f ` (s - t)\"\n    proof (rule imageE)\n      fix x\n      assume \"y = f x\"\n      assume \"x \u2208 s\"\n      have \u2039x \u2209 t\u203a\n      proof (rule notI)\n        assume \"x \u2208 t\"\n        then have \"f x \u2208 f ` t\"\n          by (rule imageI)\n        with \u2039y = f x\u203a have \"y \u2208 f ` t\"\n          by (rule ssubst)\n      with \u2039y \u2209 f ` 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 have \"f x \u2208 f ` (s - t)\"\n      by (rule imageI)\n    with \u2039y = f x\u203a show \"y \u2208 f ` (s - t)\"\n      by (rule ssubst)\n    qed\n  qed\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\n\nlemma \"f ` s - f ` t \u2286 f ` (s - t)\"\nproof\n  fix y\n  assume hy : \"y \u2208 f ` s - f ` t\"\n  then show \"y \u2208 f ` (s - t)\"\n  proof\n    assume \"y \u2208 f ` s\"\n    assume \"y \u2209 f ` t\"\n    note \u2039y \u2208 f ` s\u203a\n    then show \"y \u2208 f ` (s - t)\"\n    proof\n      fix x\n      assume \"y = f x\"\n      assume \"x \u2208 s\"\n      have \u2039x \u2209 t\u203a\n      proof\n        assume \"x \u2208 t\"\n        then have \"f x \u2208 f ` t\" by simp\n        with \u2039y = f x\u203a have \"y \u2208 f ` t\" by simp\n      with \u2039y \u2209 f ` t\u203a show False by simp\n    qed\n    with \u2039x \u2208 s\u203a have \"x \u2208 s - t\" by simp\n    then have \"f x \u2208 f ` (s - t)\" by simp\n    with \u2039y = f x\u203a show \"y \u2208 f ` (s - t)\" by simp\n    qed\n  qed\nqed\n\n(* 3\u00aa demostraci\u00f3n *)\n\nlemma \"f ` s - f ` t \u2286 f ` (s - t)\"\n  by (simp only: image_diff_subset)\n\n(* 4\u00aa demostraci\u00f3n *)\n\nlemma \"f ` s - f ` t \u2286 f ` (s - t)\"\n  by auto\n\nend\n<\/pre>\n<p><a name=\"ej10\"><\/a><\/p>\n<h3>10. f[s] \u2229 t = f[s \u2229 f\u207b\u00b9[t]]<\/h3>\n<p>Demostrar con Lean4 que<br \/>\n&#92;[ f[s] \u2229 t = f[s \u2229 f\u207b\u00b9[t]] &#92;]<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Function\nimport Mathlib.Tactic\n\nopen Set\n\nvariable {\u03b1 \u03b2 : Type _}\nvariable (f : \u03b1 \u2192 \u03b2)\nvariable (s : Set \u03b1)\nvariable (t : Set \u03b2)\n\nexample : (f '' s) \u2229 t = f '' (s \u2229 f \u207b\u00b9' t) :=\nby sorry\n<\/pre>\n<h4>10.1. Demostraci\u00f3n en lenguaje natural<\/h4>\n<p>Tenemos que demostrar que, para toda &#92;(y&#92;),<br \/>\n&#92;[ y \u2208 f[s] \u2229 t \u2194 y \u2208 f[s \u2229 f\u207b\u00b9[t]] &#92;]<br \/>\nLo haremos probando las dos implicaciones.<\/p>\n<p>(\u27f9) Supongamos que &#92;(y \u2208 f[s] \u2229 t&#92;). Entonces, se tiene que<br \/>\n&#92;begin{align}<br \/>\n   &amp;y \u2208 f[s] &#92;tag{1} &#92;&#92;<br \/>\n   &amp;y \u2208 t    &#92;tag{2}<br \/>\n&#92;end{align}<br \/>\nPor (1), existe un &#92;(x&#92;) tal que<br \/>\n&#92;begin{align}<br \/>\n   &amp;x \u2208 s     &#92;tag{3} &#92;&#92;<br \/>\n   &amp;f(x) = y  &#92;tag{4}<br \/>\n&#92;end{align}<br \/>\nPor (2) y (4),<br \/>\n&#92;[ f(x) \u2208 t &#92;]<br \/>\ny, por tanto,<br \/>\n&#92;[ x \u2208 f\u207b\u00b9[t] &#92;]<br \/>\nque, junto con (3), da<br \/>\n&#92;{ x \u2208 s \u2229 f\u207b\u00b9[t] &#92;]<br \/>\ny, por tanto,<br \/>\n&#92;[ f(x) \u2208 f[s \u2229 f\u207b\u00b9[t]] &#92;]<br \/>\nque, junto con (4), da<br \/>\n&#92;[ y \u2208 f[s \u2229 f\u207b\u00b9[t]] &#92;]<\/p>\n<p>(\u27f8) Supongamos que &#92;(y \u2208 f[s \u2229 f\u207b\u00b9[t]]&#92;). Entonces, existe un &#92;(x&#92;) tal que<br \/>\n&#92;begin{align}<br \/>\n   &amp;x \u2208 s \u2229 f\u207b\u00b9[t] &#92;tag{5} &#92;&#92;<br \/>\n   &amp;f(x) = y       &#92;tag{6}<br \/>\n&#92;end{align}<br \/>\nPor (1), se tiene que<br \/>\n&#92;begin{align}<br \/>\n   &amp;x \u2208 s      &#92;tag{7} &#92;&#92;<br \/>\n   &amp;x \u2208 f\u207b\u00b9[t] &#92;tag{8}<br \/>\n&#92;end{align}<br \/>\nPor (7) se tiene que<br \/>\n&#92;[ f(x) \u2208 f[s] &#92;]<br \/>\ny, junto con (6), se tiene que<br \/>\n&#92;[ y \u2208 f[s] &#92;tag{9} &#92;]<br \/>\nPor (8), se tiene que<br \/>\n&#92;[ f(x) \u2208 t &#92;]<br \/>\ny, junto con (6), se tiene que<br \/>\n&#92;[ y \u2208 t &#92;tag{10} &#92;]<br \/>\nPor (9) y (19), se tiene que<br \/>\n&#92;[ y \u2208 f[s] \u2229 t &#92;]<\/p>\n<h4>10.2. Demostraciones con Lean4<\/h4>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Function\nimport Mathlib.Tactic\n\nopen Set\n\nvariable {\u03b1 \u03b2 : Type _}\nvariable (f : \u03b1 \u2192 \u03b2)\nvariable (s : Set \u03b1)\nvariable (t : Set \u03b2)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (f '' s) \u2229 t = f '' (s \u2229 f \u207b\u00b9' t) :=\nby\n  ext y\n  -- y : \u03b2\n  -- \u22a2 y \u2208 f '' s \u2229 t \u2194 y \u2208 f '' (s \u2229 f \u207b\u00b9' t)\n  have h1 : y \u2208 f '' s \u2229 t \u2192 y \u2208 f '' (s \u2229 f \u207b\u00b9' t) := by\n    intro hy\n    -- hy : y \u2208 f '' s \u2229 t\n    -- \u22a2 y \u2208 f '' (s \u2229 f \u207b\u00b9' t)\n    have h1a : y \u2208 f '' s := hy.1\n    obtain \u27e8x : \u03b1, hx: x \u2208 s \u2227 f x = y\u27e9 := h1a\n    have h1b : x \u2208 s := hx.1\n    have h1c : f x = y := hx.2\n    have h1d : y \u2208 t := hy.2\n    have h1e : f x \u2208 t := by rwa [\u2190h1c] at h1d\n    have h1f : x \u2208 s \u2229 f \u207b\u00b9' t := mem_inter h1b h1e\n    have h1g : f x \u2208 f '' (s \u2229 f \u207b\u00b9' t) := mem_image_of_mem f h1f\n    show y \u2208 f '' (s \u2229 f \u207b\u00b9' t)\n    rwa [h1c] at h1g\n  have h2 : y \u2208 f '' (s \u2229 f \u207b\u00b9' t) \u2192 y \u2208 f '' s \u2229 t :=  by\n    intro hy\n    -- hy : y \u2208 f '' (s \u2229 f \u207b\u00b9' t)\n    -- \u22a2 y \u2208 f '' s \u2229 t\n    obtain \u27e8x : \u03b1, hx : x \u2208 s \u2229 f \u207b\u00b9' t \u2227 f x = y\u27e9 := hy\n    have h2a : x \u2208 s := hx.1.1\n    have h2b : f x \u2208 f '' s := mem_image_of_mem f h2a\n    have h2c : y \u2208 f '' s := by rwa [hx.2] at h2b\n    have h2d : x \u2208 f \u207b\u00b9' t := hx.1.2\n    have h2e : f x \u2208 t := mem_preimage.mp h2d\n    have h2f : y \u2208 t := by rwa [hx.2] at h2e\n    show y \u2208 f '' s \u2229 t\n    exact mem_inter h2c h2f\n  show y \u2208 f '' s \u2229 t \u2194 y \u2208 f '' (s \u2229 f \u207b\u00b9' t)\n  exact \u27e8h1, h2\u27e9\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (f '' s) \u2229 t = f '' (s \u2229 f \u207b\u00b9' t) :=\nby\n  ext y\n  -- y : \u03b2\n  -- \u22a2 y \u2208 f '' s \u2229 t \u2194 y \u2208 f '' (s \u2229 f \u207b\u00b9' t)\n  constructor\n  . -- \u22a2 y \u2208 f '' s \u2229 t \u2192 y \u2208 f '' (s \u2229 f \u207b\u00b9' t)\n    intro hy\n    -- hy : y \u2208 f '' s \u2229 t\n    -- \u22a2 y \u2208 f '' (s \u2229 f \u207b\u00b9' t)\n    cases' hy with hyfs yt\n    -- hyfs : y \u2208 f '' s\n    -- yt : y \u2208 t\n    cases' hyfs with x hx\n    -- x : \u03b1\n    -- hx : x \u2208 s \u2227 f x = y\n    cases' hx with xs fxy\n    -- xs : x \u2208 s\n    -- fxy : f x = y\n    use x\n    -- \u22a2 x \u2208 s \u2229 f \u207b\u00b9' t \u2227 f x = y\n    constructor\n    . -- \u22a2 x \u2208 s \u2229 f \u207b\u00b9' t\n      constructor\n      . -- \u22a2 x \u2208 s\n        exact xs\n      . -- \u22a2 x \u2208 f \u207b\u00b9' t\n        rw [mem_preimage]\n        -- \u22a2 f x \u2208 t\n        rw [fxy]\n        -- \u22a2 y \u2208 t\n        exact yt\n    . -- \u22a2 f x = y\n      exact fxy\n  . -- \u22a2 y \u2208 f '' (s \u2229 f \u207b\u00b9' t) \u2192 y \u2208 f '' s \u2229 t\n    intro hy\n    -- hy : y \u2208 f '' (s \u2229 f \u207b\u00b9' t)\n    -- \u22a2 y \u2208 f '' s \u2229 t\n    cases' hy with x hx\n    -- x : \u03b1\n    -- hx : x \u2208 s \u2229 f \u207b\u00b9' t \u2227 f x = y\n    constructor\n    . -- \u22a2 y \u2208 f '' s\n      use x\n      -- \u22a2 x \u2208 s \u2227 f x = y\n      constructor\n      . -- \u22a2 x \u2208 s\n        exact hx.1.1\n      . -- \u22a2 f x = y\n        exact hx.2\n    . -- \u22a2 y \u2208 t\n      cases' hx with hx1 fxy\n      -- hx1 : x \u2208 s \u2229 f \u207b\u00b9' t\n      -- fxy : f x = y\n      rw [\u2190fxy]\n      -- \u22a2 f x \u2208 t\n      rw [\u2190mem_preimage]\n      -- \u22a2 x \u2208 f \u207b\u00b9' t\n      exact hx1.2\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (f '' s) \u2229 t = f '' (s \u2229 f \u207b\u00b9' t) :=\nby\n  ext y\n  -- y : \u03b2\n  -- \u22a2 y \u2208 f '' s \u2229 t \u2194 y \u2208 f '' (s \u2229 f \u207b\u00b9' t)\n  constructor\n  . -- \u22a2 y \u2208 f '' s \u2229 t \u2192 y \u2208 f '' (s \u2229 f \u207b\u00b9' t)\n    rintro \u27e8\u27e8x, xs, fxy\u27e9, yt\u27e9\n    -- yt : y \u2208 t\n    -- x : \u03b1\n    -- xs : x \u2208 s\n    -- fxy : f x = y\n    -- \u22a2 y \u2208 f '' (s \u2229 f \u207b\u00b9' t)\n    use x\n    -- \u22a2 x \u2208 s \u2229 f \u207b\u00b9' t \u2227 f x = y\n    constructor\n    . -- \u22a2 x \u2208 s \u2229 f \u207b\u00b9' t\n      constructor\n      . -- \u22a2 x \u2208 s\n        exact xs\n      . -- \u22a2 x \u2208 f \u207b\u00b9' t\n        rw [mem_preimage]\n        -- \u22a2 f x \u2208 t\n        rw [fxy]\n        -- \u22a2 y \u2208 t\n        exact yt\n    . -- \u22a2 f x = y\n      exact fxy\n  . -- \u22a2 y \u2208 f '' (s \u2229 f \u207b\u00b9' t) \u2192 y \u2208 f '' s \u2229 t\n    rintro \u27e8x, \u27e8xs, xt\u27e9, fxy\u27e9\n    -- x : \u03b1\n    -- fxy : f x = y\n    -- xs : x \u2208 s\n    -- xt : x \u2208 f \u207b\u00b9' t\n    -- \u22a2 y \u2208 f '' s \u2229 t\n    constructor\n    . -- \u22a2 y \u2208 f '' s\n      use x, xs\n      -- \u22a2 f x = y\n      exact fxy\n    . -- \u22a2 y \u2208 t\n      rw [\u2190fxy]\n      -- \u22a2 f x \u2208 t\n      rw [\u2190mem_preimage]\n      -- \u22a2 x \u2208 f \u207b\u00b9' t\n      exact xt\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (f '' s) \u2229 t = f '' (s \u2229 f \u207b\u00b9' t) :=\nby\n  ext y\n  -- y : \u03b2\n  -- \u22a2 y \u2208 f '' s \u2229 t \u2194 y \u2208 f '' (s \u2229 f \u207b\u00b9' t)\n  constructor\n  . -- \u22a2 y \u2208 f '' s \u2229 t \u2192 y \u2208 f '' (s \u2229 f \u207b\u00b9' t)\n    rintro \u27e8\u27e8x, xs, fxy\u27e9, yt\u27e9\n    -- yt : y \u2208 t\n    -- x : \u03b1\n    -- xs : x \u2208 s\n    -- fxy : f x = y\n    -- \u22a2 y \u2208 f '' (s \u2229 f \u207b\u00b9' t)\n    aesop\n  . -- \u22a2 y \u2208 f '' (s \u2229 f \u207b\u00b9' t) \u2192 y \u2208 f '' s \u2229 t\n    rintro \u27e8x, \u27e8xs, xt\u27e9, fxy\u27e9\n    -- x : \u03b1\n    -- fxy : f x = y\n    -- xs : x \u2208 s\n    -- xt : x \u2208 f \u207b\u00b9' t\n    -- \u22a2 y \u2208 f '' s \u2229 t\n    aesop\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (f '' s) \u2229 t = f '' (s \u2229 f \u207b\u00b9' t) :=\nby ext ; constructor <;> aesop\n\n-- 6\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (f '' s) \u2229 t = f '' (s \u2229 f \u207b\u00b9' t) :=\n(image_inter_preimage f s t).symm\n\n-- Lemas usados\n-- ============\n\n-- variable (x : \u03b1)\n-- variable (v : Set \u03b1)\n-- #check (image_inter_preimage f s t : f '' (s \u2229 f \u207b\u00b9' t) = f '' s \u2229 t)\n-- #check (mem_image_of_mem f : x \u2208 s \u2192 f x \u2208 f '' s)\n-- #check (mem_inter : x \u2208 s \u2192 x \u2208 v \u2192 x \u2208 s \u2229 v)\n-- #check (mem_preimage : x \u2208 f \u207b\u00b9' t \u2194 f x \u2208 t)\n<\/pre>\n<p>Se puede interactuar con las demostraciones anteriores en<br \/>\n<a href=\"https:\/\/live.lean-lang.org\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus2\/main\/src\/Interseccion_con_la_imagen.lean\">Lean 4 Web<\/a>.<\/p>\n<h4>10.3. Demostraciones con Isabelle\/HOL<\/h4>\n<pre lang=\"isar\">\ntheory Interseccion_con_la_imagen\nimports Main\nbegin\n\n(* 1\u00aa demostraci\u00f3n *)\n\nlemma \"(f ` s) \u2229 v = f ` (s \u2229 f -` v)\"\nproof (rule equalityI)\n  show \"(f ` s) \u2229 v \u2286 f ` (s \u2229 f -` v)\"\n  proof (rule subsetI)\n    fix y\n    assume \"y \u2208 (f ` s) \u2229 v\"\n    then show \"y \u2208 f ` (s \u2229 f -` v)\"\n    proof (rule IntE)\n      assume \"y \u2208 v\"\n      assume \"y \u2208 f ` s\"\n      then show \"y \u2208 f ` (s \u2229 f -` v)\"\n      proof (rule imageE)\n        fix x\n        assume \"x \u2208 s\"\n        assume \"y = f x\"\n        then have \"f x \u2208 v\"\n          using \u2039y \u2208 v\u203a by (rule subst)\n        then have \"x \u2208 f -` v\"\n          by (rule vimageI2)\n        with \u2039x \u2208 s\u203a have \"x \u2208 s \u2229 f -` v\"\n          by (rule IntI)\n        then have \"f x \u2208 f ` (s \u2229 f -` v)\"\n          by (rule imageI)\n        with \u2039y = f x\u203a show \"y \u2208 f ` (s \u2229 f -` v)\"\n          by (rule ssubst)\n      qed\n    qed\n  qed\nnext\n  show \"f ` (s \u2229 f -` v) \u2286 (f ` s) \u2229 v\"\n  proof (rule subsetI)\n    fix y\n    assume \"y \u2208 f ` (s \u2229 f -` v)\"\n    then show \"y \u2208 (f ` s) \u2229 v\"\n    proof (rule imageE)\n      fix x\n      assume \"y = f x\"\n      assume hx : \"x \u2208 s \u2229 f -` v\"\n      have \"y \u2208 f ` s\"\n      proof -\n        have \"x \u2208 s\"\n          using hx by (rule IntD1)\n        then have \"f x \u2208 f ` s\"\n          by (rule imageI)\n        with \u2039y = f x\u203a show \"y \u2208 f ` s\"\n          by (rule ssubst)\n      qed\n      moreover\n      have \"y \u2208 v\"\n      proof -\n        have \"x \u2208 f -` v\"\n          using hx by (rule IntD2)\n        then have \"f x \u2208 v\"\n          by (rule vimageD)\n        with \u2039y = f x\u203a show \"y \u2208 v\"\n          by (rule ssubst)\n      qed\n      ultimately show \"y \u2208 (f ` s) \u2229 v\"\n        by (rule IntI)\n    qed\n  qed\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\n\nlemma \"(f ` s) \u2229 v = f ` (s \u2229 f -` v)\"\nproof\n  show \"(f ` s) \u2229 v \u2286 f ` (s \u2229 f -` v)\"\n  proof\n    fix y\n    assume \"y \u2208 (f ` s) \u2229 v\"\n    then show \"y \u2208 f ` (s \u2229 f -` v)\"\n    proof\n      assume \"y \u2208 v\"\n      assume \"y \u2208 f ` s\"\n      then show \"y \u2208 f ` (s \u2229 f -` v)\"\n      proof\n        fix x\n        assume \"x \u2208 s\"\n        assume \"y = f x\"\n        then have \"f x \u2208 v\" using \u2039y \u2208 v\u203a by simp\n        then have \"x \u2208 f -` v\" by simp\n        with \u2039x \u2208 s\u203a have \"x \u2208 s \u2229 f -` v\" by simp\n        then have \"f x \u2208 f ` (s \u2229 f -` v)\" by simp\n        with \u2039y = f x\u203a show \"y \u2208 f ` (s \u2229 f -` v)\" by simp\n      qed\n    qed\n  qed\nnext\n  show \"f ` (s \u2229 f -` v) \u2286 (f ` s) \u2229 v\"\n  proof\n    fix y\n    assume \"y \u2208 f ` (s \u2229 f -` v)\"\n    then show \"y \u2208 (f ` s) \u2229 v\"\n    proof\n      fix x\n      assume \"y = f x\"\n      assume hx : \"x \u2208 s \u2229 f -` v\"\n      have \"y \u2208 f ` s\"\n      proof -\n        have \"x \u2208 s\" using hx by simp\n        then have \"f x \u2208 f ` s\" by simp\n        with \u2039y = f x\u203a show \"y \u2208 f ` s\" by simp\n      qed\n      moreover\n      have \"y \u2208 v\"\n      proof -\n        have \"x \u2208 f -` v\" using hx by simp\n        then have \"f x \u2208 v\" by simp\n        with \u2039y = f x\u203a show \"y \u2208 v\" by simp\n      qed\n      ultimately show \"y \u2208 (f ` s) \u2229 v\" by simp\n    qed\n  qed\nqed\n\n(* 3\u00aa demostraci\u00f3n *)\n\nlemma \"(f ` s) \u2229 v = f ` (s \u2229 f -` v)\"\nproof\n  show \"(f ` s) \u2229 v \u2286 f ` (s \u2229 f -` v)\"\n  proof\n    fix y\n    assume \"y \u2208 (f ` s) \u2229 v\"\n    then show \"y \u2208 f ` (s \u2229 f -` v)\"\n    proof\n      assume \"y \u2208 v\"\n      assume \"y \u2208 f ` s\"\n      then show \"y \u2208 f ` (s \u2229 f -` v)\"\n      proof\n        fix x\n        assume \"x \u2208 s\"\n        assume \"y = f x\"\n        then show \"y \u2208 f ` (s \u2229 f -` v)\"\n          using \u2039x \u2208 s\u203a \u2039y \u2208 v\u203a by simp\n      qed\n    qed\n  qed\nnext\n  show \"f ` (s \u2229 f -` v) \u2286 (f ` s) \u2229 v\"\n  proof\n    fix y\n    assume \"y \u2208 f ` (s \u2229 f -` v)\"\n    then show \"y \u2208 (f ` s) \u2229 v\"\n    proof\n      fix x\n      assume \"y = f x\"\n      assume hx : \"x \u2208 s \u2229 f -` v\"\n      then have \"y \u2208 f ` s\" using \u2039y = f x\u203a by simp\n      moreover\n      have \"y \u2208 v\" using hx \u2039y = f x\u203a by simp\n      ultimately show \"y \u2208 (f ` s) \u2229 v\" by simp\n    qed\n  qed\nqed\n\n(* 4\u00aa demostraci\u00f3n *)\n\nlemma \"(f ` s) \u2229 v = f ` (s \u2229 f -` v)\"\n  by auto\n\nend\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Desde el 18 de marzo, he publicado en Calculemus las demostraciones con Lean4 e Isabelle\/HOL de las siguientes propiedades: 1. Si f es inyectiva, entonces f\u207b\u00b9(f(s)\u200b) \u2286 s 2. f(f\u207b\u00b9(u)) \u2286 u 3. Si f es suprayectiva, entonces u \u2286 f(f\u207b\u00b9(u)) 4. Si s \u2286 t, entonces f(s) \u2286 f(t) 5. Si u \u2286 v,&#8230;<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_kad_post_transparent":"default","_kad_post_title":"default","_kad_post_layout":"default","_kad_post_sidebar_id":"","_kad_post_content_style":"default","_kad_post_vertical_padding":"default","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"footnotes":"","_jetpack_memberships_contains_paid_content":false},"categories":[1],"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\/8174"}],"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=8174"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/8174\/revisions"}],"predecessor-version":[{"id":8177,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/8174\/revisions\/8177"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/media?parent=8174"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/categories?post=8174"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/tags?post=8174"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}