{"id":7443,"date":"2020-10-03T17:25:34","date_gmt":"2020-10-03T15:25:34","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/?p=7443"},"modified":"2020-12-20T17:26:19","modified_gmt":"2020-12-20T16:26:19","slug":"formatus-pruebas-en-lean-de-%c2%ac%e2%88%80x-px-%e2%86%94-%e2%88%83x-%c2%acpx","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/formatus-pruebas-en-lean-de-%c2%ac%e2%88%80x-px-%e2%86%94-%e2%88%83x-%c2%acpx\/","title":{"rendered":"ForMatUS: Pruebas en Lean de \u00ac\u2200x P(x) \u2194 \u2203x \u00acP(x)"},"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\/ldZfF6P5pNs\">v\u00eddeo<\/a> en el que se comentan pruebas en Lean de la propiedad<\/p>\n<pre lang=\"lean\">\u00ac\u2200x P(x) \u2194 \u2203x \u00acP(x)\n<\/pre>\n<p>usando los estilos declarativos, aplicativos, funcional y autom\u00e1tico.<\/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\/ldZfF6P5pNs\" 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\/2_LPO\/Pruebas_de_%C2%AC%E2%88%80xP(x)%E2%86%94%E2%88%83x%C2%ACP(x).lean\">c\u00f3digo<\/a> de la teor\u00eda utilizada<\/p>\n<pre lang=\"lean\">import tactic\n\nvariable {U : Type}\nvariable {P : U -&gt; Prop}\n\n-- ------------------------------------------------------\n-- Ej. 1. Demostrar que\n--    \u00ac\u2200x P(x) \u22a2 \u2203x \u00acP(x)\n-- ------------------------------------------------------\n\n-- 1\u00aa demostraci\u00f3n\nexample\n  (h1 : \u00ac\u2200x, P x)\n  : \u2203x, \u00acP x :=\nby_contra\n  ( assume h2 : \u00ac\u2203x, \u00acP x,\n    have h8 : \u2200x, P x, from\n      ( assume x\u2080,\n        show P x\u2080, from\n          by_contra\n            ( assume h4 : \u00acP x\u2080,\n              have h5 : \u2203x, \u00acP x, from exists.intro x\u2080 h4,\n              show false, from h2 h5 )),\n    show false, from h1 h8)\n\n-- 2\u00aa demostraci\u00f3n\nexample\n  (h1 : \u00ac\u2200x, P x)\n  : \u2203x, \u00acP x :=\nby_contra\n  ( assume h2 : \u00ac\u2203x, \u00acP x,\n    have h8 : \u2200x, P x, from\n      ( assume x\u2080,\n        show P x\u2080, from\n          by_contra\n            ( assume h4 : \u00acP x\u2080,\n              have h5 : \u2203x, \u00acP x, from exists.intro x\u2080 h4,\n              show false, from h2 h5 )),\n    h1 h8)\n\n-- 3\u00aa demostraci\u00f3n\nexample\n  (h1 : \u00ac\u2200x, P x)\n  : \u2203x, \u00acP x :=\nby_contra\n  ( assume h2 : \u00ac\u2203x, \u00acP x,\n    have h8 : \u2200x, P x, from\n      ( assume x\u2080,\n        show P x\u2080, from\n          by_contra\n            ( assume h4 : \u00acP x\u2080,\n              have h5 : \u2203x, \u00acP x, from exists.intro x\u2080 h4,\n              h2 h5 )),\n    h1 h8)\n\n-- 4\u00aa demostraci\u00f3n\nexample\n  (h1 : \u00ac\u2200x, P x)\n  : \u2203x, \u00acP x :=\nby_contra\n  ( assume h2 : \u00ac\u2203x, \u00acP x,\n    have h8 : \u2200x, P x, from\n      ( assume x\u2080,\n        show P x\u2080, from\n          by_contra\n            ( assume h4 : \u00acP x\u2080,\n              have h5 : \u2203x, \u00acP x, from \u27e8x\u2080, h4\u27e9,\n              h2 h5 )),\n    h1 h8)\n\n-- 5\u00aa demostraci\u00f3n\nexample\n  (h1 : \u00ac\u2200x, P x)\n  : \u2203x, \u00acP x :=\nby_contra\n  ( assume h2 : \u00ac\u2203x, \u00acP x,\n    have h8 : \u2200x, P x, from\n      ( assume x\u2080,\n        show P x\u2080, from\n          by_contra\n            ( assume h4 : \u00acP x\u2080,\n              h2 \u27e8x\u2080, h4\u27e9 )),\n    h1 h8)\n\n-- 6\u00aa demostraci\u00f3n\nexample\n  (h1 : \u00ac\u2200x, P x)\n  : \u2203x, \u00acP x :=\nby_contra\n  ( assume h2 : \u00ac\u2203x, \u00acP x,\n    have h8 : \u2200x, P x, from\n      ( assume x\u2080,\n        show P x\u2080, from\n          by_contra (\u03bb h4, h2 \u27e8x\u2080, h4\u27e9)),\n    h1 h8)\n\n-- 7\u00aa demostraci\u00f3n\nexample\n  (h1 : \u00ac\u2200x, P x)\n  : \u2203x, \u00acP x :=\nby_contra\n  ( assume h2 : \u00ac\u2203x, \u00acP x,\n    have h8 : \u2200x, P x, from\n      ( assume x\u2080,\n        by_contra (\u03bb h4, h2 \u27e8x\u2080, h4\u27e9)),\n    h1 h8)\n\n-- 8\u00aa demostraci\u00f3n\nexample\n  (h1 : \u00ac\u2200x, P x)\n  : \u2203x, \u00acP x :=\nby_contra\n  ( assume h2 : \u00ac\u2203x, \u00acP x,\n    have h8 : \u2200x, P x, from\n      (\u03bb x\u2080, by_contra (\u03bb h4, h2 \u27e8x\u2080, h4\u27e9)),\n    h1 h8)\n\n-- 9\u00aa demostraci\u00f3n\nexample\n  (h1 : \u00ac\u2200x, P x)\n  : \u2203x, \u00acP x :=\nby_contra\n  ( assume h2 : \u00ac\u2203x, \u00acP x,\n    h1 (\u03bb x\u2080, by_contra (\u03bb h4, h2 \u27e8x\u2080, h4\u27e9)))\n\n-- 10\u00aa demostraci\u00f3n\nexample\n  (h1 : \u00ac\u2200x, P x)\n  : \u2203x, \u00acP x :=\nby_contra (\u03bb h2, h1 (\u03bb x\u2080, by_contra (\u03bb h4, h2 \u27e8x\u2080, h4\u27e9)))\n\n-- 11\u00aa demostraci\u00f3n\nexample\n  (h1 : \u00ac\u2200x, P x)\n  : \u2203x, \u00acP x :=\n-- by library_search\nnot_forall.mp h1\n\n-- 12\u00aa demostraci\u00f3n\nlemma aux1\n  (h1 : \u00ac\u2200x, P x)\n  : \u2203x, \u00acP x :=\n-- by hint\nby finish\n\n-- ------------------------------------------------------\n-- Ej. 2. Demostrar que\n--    \u2203x \u00acP(x) \u22a2 \u00ac\u2200x P(x)\n-- ------------------------------------------------------\n\n-- 1\u00aa demostraci\u00f3n\nexample\n  (h1 : \u2203x, \u00acP x)\n  : \u00ac\u2200x, P x :=\nassume h2 : \u2200x, P x,\nexists.elim h1\n  ( assume x\u2080 (h3 : \u00acP x\u2080),\n    have h4 : P x\u2080, from h2 x\u2080,\n    show false, from h3 h4)\n\n-- 2\u00aa demostraci\u00f3n\nexample\n  (h1 : \u2203x, \u00acP x)\n  : \u00ac\u2200x, P x :=\nassume h2 : \u2200x, P x,\nexists.elim h1\n  ( assume x\u2080 (h3 : \u00acP x\u2080),\n    have h4 : P x\u2080, from h2 x\u2080,\n    h3 h4)\n\n-- 3\u00aa demostraci\u00f3n\nexample\n  (h1 : \u2203x, \u00acP x)\n  : \u00ac\u2200x, P x :=\nassume h2 : \u2200x, P x,\nexists.elim h1\n  ( assume x\u2080 (h3 : \u00acP x\u2080),\n    h3 (h2 x\u2080))\n\n-- 4\u00aa demostraci\u00f3n\nexample\n  (h1 : \u2203x, \u00acP x)\n  : \u00ac\u2200x, P x :=\nassume h2 : \u2200x, P x,\nexists.elim h1\n  (\u03bb x\u2080 h3, h3 (h2 x\u2080))\n\n-- 5\u00aa demostraci\u00f3n\nexample\n  (h1 : \u2203x, \u00acP x)\n  : \u00ac\u2200x, P x :=\n\u03bb h2, exists.elim h1 (\u03bb x\u2080 h3, h3 (h2 x\u2080))\n\n-- 6\u00aa demostraci\u00f3n\nexample\n  (h1 : \u2203x, \u00acP x)\n  : \u00ac\u2200x, P x :=\n-- by library_search\nnot_forall.mpr h1\n\n-- 7\u00aa demostraci\u00f3n\nexample\n  (h1 : \u2203x, \u00acP x)\n  : \u00ac\u2200x, P x :=\nassume h2 : \u2200x, P x,\nmatch h1 with \u27e8x\u2080, (h3 : \u00acP x\u2080)\u27e9 :=\n  ( have h4 : P x\u2080, from h2 x\u2080,\n    show false,     from h3 h4)\nend\n\n-- 8\u00aa demostraci\u00f3n\nexample\n  (h1 : \u2203x, \u00acP x)\n  : \u00ac\u2200x, P x :=\nbegin\n  intro h2,\n  cases h1 with x\u2080 h3,\n  apply h3,\n  apply h2,\nend\n\nexample\n  (h1 : \u2203x, \u00acP x)\n  : \u00ac\u2200x, P x :=\nbegin\n  intro h2,\n  obtain \u27e8x\u2080, h3\u27e9 := h1,\n  apply h3,\n  apply h2,\nend\n\n-- 9\u00aa demostraci\u00f3n\nexample\n  (h1 : \u2203x, \u00acP x)\n  : \u00ac\u2200x, P x :=\n-- by hint\nby tauto\n\n-- 10\u00aa demostraci\u00f3n\nlemma aux2\n  (h1 : \u2203x, \u00acP x)\n  : \u00ac\u2200x, P x :=\nby finish\n\n#print axioms aux2\n\n-- ------------------------------------------------------\n-- Ej. 3. Demostrar que\n--    \u00ac\u2200x P(x) \u2194 \u2203x \u00acP(x)\n-- ------------------------------------------------------\n\n-- 1\u00aa demostraci\u00f3n\nexample :\n  (\u00ac\u2200x, P x) \u2194 (\u2203x, \u00acP x) :=\niff.intro\n  ( assume h1 : \u00ac\u2200x, P x,\n    show \u2203x, \u00acP x, from aux1 h1)\n  ( assume h2 : \u2203x, \u00acP x,\n    show \u00ac\u2200x, P x, from aux2 h2)\n\n-- 2\u00aa demostraci\u00f3n\nexample :\n  (\u00ac\u2200x, P x) \u2194 (\u2203x, \u00acP x) :=\niff.intro aux1 aux2\n\n-- 3\u00aa demostraci\u00f3n\nexample :\n  (\u00ac\u2200x, P x) \u2194 (\u2203x, \u00acP x) :=\n-- by library_search\nnot_forall\n\n-- 4\u00aa demostraci\u00f3n\nexample :\n  (\u00ac\u2200x, P x) \u2194 (\u2203x, \u00acP x) :=\nbegin\n  split,\n  { exact aux1, },\n  { exact aux2, },\nend\n\n-- 5\u00aa demostraci\u00f3n\nexample :\n  (\u00ac\u2200x, P x) \u2194 (\u2203x, \u00acP x) :=\n-- by hint\nby finish\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 la propiedad \u00ac\u2200x P(x) \u2194 \u2203x \u00acP(x) usando los estilos declarativos, aplicativos, funcional y autom\u00e1tico. A continuaci\u00f3n, se muestra el v\u00eddeo y el c\u00f3digo de la teor\u00eda utilizada import tactic variable {U : Type} variable {P&#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":[336],"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\/7443"}],"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=7443"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7443\/revisions"}],"predecessor-version":[{"id":7444,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7443\/revisions\/7444"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/media?parent=7443"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/categories?post=7443"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/tags?post=7443"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}