{"id":7523,"date":"2020-11-03T19:16:56","date_gmt":"2020-11-03T18:16:56","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/?p=7523"},"modified":"2020-12-21T19:17:48","modified_gmt":"2020-12-21T18:17:48","slug":"formatus-pruebas-en-lean-de-propiedades-de-la-composicion-de-funciones-elemento-neutro-y-asociatividad","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/formatus-pruebas-en-lean-de-propiedades-de-la-composicion-de-funciones-elemento-neutro-y-asociatividad\/","title":{"rendered":"ForMatUS: Pruebas en Lean de propiedades de la composici\u00f3n de funciones (elemento neutro y asociatividad)"},"content":{"rendered":"<p>He a\u00f1adido a la lista <a href=\"https:\/\/bit.ly\/2FcUrwQ\">L\u00f3gica con Lean<\/a> el <a href=\"https:\/\/youtu.be\/6-VglWN0e7g\">v\u00eddeo<\/a> en el que se comentan pruebas en Lean de propiedades de la composici\u00f3n de funciones:<\/p>\n<ul class=\"org-ul\">\n<li>la funci\u00f3n identidad es el elemento neutro de la composici\u00f3n y<\/li>\n<li>la composici\u00f3n es asociativa.<\/li>\n<\/ul>\n<p>En las pruebas se usan los estilos aplicativos y declarativos.<\/p>\n<p>A continuaci\u00f3n, se muestra el v\u00eddeo<\/p>\n<p><center><\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/6-VglWN0e7g\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<p><\/center>y el <a href=\"https:\/\/github.com\/jaalonso\/Logica_con_Lean\/blob\/master\/src\/5_Funciones\/Propiedades_de_la_composicion_de_funciones.lean\">c\u00f3digo<\/a> de la teor\u00eda utilizada<\/p>\n<pre lang=\"lean\">import tactic\n\nopen function\n\nvariables {X Y Z W : Type}\n\n-- ----------------------------------------------------\n-- Ej. 1. Demostrar que\n--    id \u2218 f = f\n-- ----------------------------------------------------\n\n-- 1\u00aa demostraci\u00f3n\nexample\n  (f : X \u2192 Y)\n  : id \u2218 f = f :=\nbegin\n  ext,\n  calc (id \u2218 f) x = id (f x) : by rw comp_app\n       ...        = f x      : by rw id.def,\nend\n\n-- 2\u00aa demostraci\u00f3n\nexample\n  (f : X \u2192 Y)\n  : id \u2218 f = f :=\nbegin\n  ext,\n  rw comp_app,\n  rw id.def,\nend\n\n-- 3\u00aa demostraci\u00f3n\nexample\n  (f : X \u2192 Y)\n  : id \u2218 f = f :=\nbegin\n  ext,\n  rw [comp_app, id.def],\nend\n\n-- 4\u00aa demostraci\u00f3n\nexample\n  (f : X \u2192 Y)\n  : id \u2218 f = f :=\nbegin\n  ext,\n  calc (id \u2218 f) x = id (f x) : rfl\n       ...        = f x      : rfl,\nend\n\n-- 5\u00aa demostraci\u00f3n\nexample\n  (f : X \u2192 Y)\n  : id \u2218 f = f :=\nrfl\n\n-- 6\u00aa demostraci\u00f3n\nexample\n  (f : X \u2192 Y)\n  : id \u2218 f = f :=\n-- by library_search\nleft_id f\n\n-- 7\u00aa demostraci\u00f3n\nexample\n  (f : X \u2192 Y)\n  : id \u2218 f = f :=\ncomp.left_id f\n\n-- ----------------------------------------------------\n-- Ej. 2. Demostrar que\n--    f \u2218 id = f\n-- ----------------------------------------------------\n\n-- 1\u00aa demostraci\u00f3n\nexample\n  (f : X \u2192 Y)\n  : f \u2218 id = f :=\nbegin\n  ext,\n  calc (f \u2218 id) x = f (id x) : by rw comp_app\n       ...        = f x      : by rw id.def,\nend\n\n-- 2\u00aa demostraci\u00f3n\nexample\n  (f : X \u2192 Y)\n  : f \u2218 id = f :=\nbegin\n  ext,\n  rw comp_app,\n  rw id.def,\nend\n\n-- 3\u00aa demostraci\u00f3n\nexample\n  (f : X \u2192 Y)\n  : f \u2218 id = f :=\nbegin\n  ext,\n  rw [comp_app, id.def],\nend\n\n-- 4\u00aa demostraci\u00f3n\nexample\n  (f : X \u2192 Y)\n  : f \u2218 id = f :=\nbegin\n  ext,\n  calc (f \u2218 id) x = f (id x) : rfl\n       ...        = f x      : rfl,\nend\n\n-- 5\u00aa demostraci\u00f3n\nexample\n  (f : X \u2192 Y)\n  : f \u2218 id = f :=\nrfl\n\n-- 6\u00aa demostraci\u00f3n\nexample\n  (f : X \u2192 Y)\n  : f \u2218 id = f :=\n-- by library_search\nright_id f\n\n-- 7\u00aa demostraci\u00f3n\nexample\n  (f : X \u2192 Y)\n  : f \u2218 id = f :=\ncomp.right_id f\n\n-- ----------------------------------------------------\n-- Ej. 3. Demostrar que\n--    (f \u2218 g) \u2218 h = f \u2218 (g \u2218 h)\n-- ----------------------------------------------------\n\n-- 1\u00aa demostraci\u00f3n\nexample\n  (f : Z \u2192 W)\n  (g : Y \u2192 Z)\n  (h : X \u2192 Y)\n  : (f \u2218 g) \u2218 h = f \u2218 (g \u2218 h) :=\nbegin\n  ext,\n  calc ((f \u2218 g) \u2218 h) x\n           = (f \u2218 g) (h x)   : by rw comp_app\n       ... = f (g (h x))     : by rw comp_app\n       ... = f ((g \u2218 h) x)   : by rw comp_app\n       ... = (f \u2218 (g \u2218 h)) x : by rw comp_app\nend\n\n-- 2\u00aa demostraci\u00f3n\nexample\n  (f : Z \u2192 W)\n  (g : Y \u2192 Z)\n  (h : X \u2192 Y)\n  : (f \u2218 g) \u2218 h = f \u2218 (g \u2218 h) :=\nbegin\n  ext,\n  rw comp_app,\nend\n\n-- 3\u00aa demostraci\u00f3n\nexample\n  (f : Z \u2192 W)\n  (g : Y \u2192 Z)\n  (h : X \u2192 Y)\n  : (f \u2218 g) \u2218 h = f \u2218 (g \u2218 h) :=\nbegin\n  ext,\n  calc ((f \u2218 g) \u2218 h) x\n           = (f \u2218 g) (h x)   : rfl\n       ... = f (g (h x))     : rfl\n       ... = f ((g \u2218 h) x)   : rfl\n       ... = (f \u2218 (g \u2218 h)) x : rfl\nend\n\n-- 4\u00aa demostraci\u00f3n\nexample\n  (f : Z \u2192 W)\n  (g : Y \u2192 Z)\n  (h : X \u2192 Y)\n  : (f \u2218 g) \u2218 h = f \u2218 (g \u2218 h) :=\nrfl\n\n-- 5\u00aa demostraci\u00f3n\nexample\n  (f : Z \u2192 W)\n  (g : Y \u2192 Z)\n  (h : X \u2192 Y)\n  : (f \u2218 g) \u2218 h = f \u2218 (g \u2218 h) :=\ncomp.assoc f g h\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>He a\u00f1adido a la lista L\u00f3gica con Lean el v\u00eddeo en el que se comentan pruebas en Lean de propiedades de la composici\u00f3n de funciones: la funci\u00f3n identidad es el elemento neutro de la composici\u00f3n y la composici\u00f3n es asociativa. En las pruebas se usan los estilos aplicativos y declarativos. A continuaci\u00f3n, se muestra el&#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":"","_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":[166,339],"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\/7523"}],"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=7523"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7523\/revisions"}],"predecessor-version":[{"id":7524,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7523\/revisions\/7524"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/media?parent=7523"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/categories?post=7523"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/tags?post=7523"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}