{"id":7427,"date":"2020-09-30T13:52:20","date_gmt":"2020-09-30T11:52:20","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/?p=7427"},"modified":"2020-12-20T13:53:23","modified_gmt":"2020-12-20T12:53:23","slug":"formatus-pruebas-en-lean-de-p-%e2%86%92-q-%e2%8a%a2-%c2%acp-%e2%88%a8-q","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/formatus-pruebas-en-lean-de-p-%e2%86%92-q-%e2%8a%a2-%c2%acp-%e2%88%a8-q\/","title":{"rendered":"ForMatUS: Pruebas en Lean de P \u2192 Q \u22a2 \u00acP \u2228 Q"},"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\/1_3Th-HOARI\">v\u00eddeo<\/a> en el que se comentan 13 pruebas en Lean de la propiedad<\/p>\n<pre lang=\"lean\">P \u2192 Q \u22a2 \u00acP \u2228 Q\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\/1_3Th-HOARI\" 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\/1_Proposicional\/Pruebas_de_P%E2%86%92Q%E2%8A%A2%C2%ACP%E2%88%A8Q.lean\">c\u00f3digo<\/a> de la teor\u00eda utilizada<\/p>\n<pre lang=\"lean\">-- ----------------------------------------------------\n-- Ej. 1. (p. 24) Demostrar\n--    P \u2192 Q \u22a2 \u00acP \u2228 Q\n-- ----------------------------------------------------\n\nimport tactic\n\nvariables (P Q : Prop)\n\nopen_locale classical\n\n-- 1\u00aa demostraci\u00f3n\nexample\n  (h1 : P \u2192 Q)\n  : \u00acP \u2228 Q :=\nhave h2 : P \u2228 \u00acP,\n  from em P,\nor.elim h2\n  ( assume h3 : P,\n    have h4 : Q,\n      from h1 h3,\n    show \u00acP \u2228 Q,\n      from or.inr h4)\n  ( assume h5 : \u00acP,\n    show \u00acP \u2228 Q,\n      from or.inl h5)\n\n-- 2\u00aa demostraci\u00f3n\nexample\n  (h1 : P \u2192 Q)\n  : \u00acP \u2228 Q :=\nhave h2 : P \u2228 \u00acP,\n  from em P,\nor.elim h2\n  ( assume h3 : P,\n    have h4 : Q,\n      from h1 h3,\n    show \u00acP \u2228 Q,\n      from or.inr h4)\n  ( assume h5 : \u00acP,\n    or.inl h5)\n\n-- 3\u00aa demostraci\u00f3n\nexample\n  (h1 : P \u2192 Q)\n  : \u00acP \u2228 Q :=\nhave h2 : P \u2228 \u00acP,\n  from em P,\nor.elim h2\n  ( assume h3 : P,\n    have h4 : Q,\n      from h1 h3,\n    show \u00acP \u2228 Q,\n      from or.inr h4)\n  ( \u03bb h5, or.inl h5)\n\n-- 4\u00aa demostraci\u00f3n\nexample\n  (h1 : P \u2192 Q)\n  : \u00acP \u2228 Q :=\nhave h2 : P \u2228 \u00acP,\n  from em P,\nor.elim h2\n  ( assume h3 : P,\n    have h4 : Q,\n      from h1 h3,\n    or.inr h4)\n  ( \u03bb h5, or.inl h5)\n\n-- 5\u00aa demostraci\u00f3n\nexample\n  (h1 : P \u2192 Q)\n  : \u00acP \u2228 Q :=\nhave h2 : P \u2228 \u00acP,\n  from em P,\nor.elim h2\n  ( assume h3 : P,\n    or.inr (h1 h3))\n  ( \u03bb h5, or.inl h5)\n\n-- 6\u00aa demostraci\u00f3n\nexample\n  (h1 : P \u2192 Q)\n  : \u00acP \u2228 Q :=\nhave h2 : P \u2228 \u00acP,\n  from em P,\nor.elim h2\n  ( \u03bb h3, or.inr (h1 h3))\n  ( \u03bb h5, or.inl h5)\n\n-- 7\u00aa demostraci\u00f3n\nexample\n  (h1 : P \u2192 Q)\n  : \u00acP \u2228 Q :=\nor.elim (em P)\n  ( \u03bb h3, or.inr (h1 h3))\n  ( \u03bb h5, or.inl h5)\n\nexample\n  (h1 : P \u2192 Q)\n  : \u00acP \u2228 Q :=\n(em P).elim\n  ( \u03bb h3, or.inr (h1 h3))\n  ( \u03bb h5, or.inl h5)\n\n-- 8\u00aa demostraci\u00f3n\nexample\n  (h1 : P \u2192 Q)\n  : \u00acP \u2228 Q :=\n-- by library_search\nnot_or_of_imp h1\n\n-- 9\u00aa demostraci\u00f3n\nexample\n  (h1 : P \u2192 Q)\n  : \u00acP \u2228 Q :=\nif h3 : P then or.inr (h1 h3) else or.inl h3\n\n-- 10\u00aa demostraci\u00f3n\nexample\n  (h1 : P \u2192 Q)\n  : \u00acP \u2228 Q :=\nbegin\n  by_cases h2 : P,\n  { apply or.inr,\n    exact h1 h2, },\n  { exact or.inl h2, },\nend\n\n-- 11\u00aa demostraci\u00f3n\nexample\n  (h1 : P \u2192 Q)\n  : \u00acP \u2228 Q :=\nbegin\n  by_cases h2 : P,\n  { exact or.inr (h1 h2), },\n  { exact or.inl h2, },\nend\n\n-- 12\u00aa demostraci\u00f3n\nexample\n  (h1 : P \u2192 Q)\n  : \u00acP \u2228 Q :=\nbegin\n  by_cases h2 : P,\n  { right,\n    exact h1 h2, },\n  { left,\n    exact h2, },\nend\n\n-- 13\u00aa demostraci\u00f3n\nexample\n  (h1 : P \u2192 Q)\n  : \u00acP \u2228 Q :=\n-- by hint\nby tauto\n\n-- 14\u00aa demostraci\u00f3n\nexample\n  (h1 : P \u2192 Q)\n  : \u00acP \u2228 Q :=\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 13 pruebas en Lean de la propiedad P \u2192 Q \u22a2 \u00acP \u2228 Q 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 &#8212; &#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;- &#8212; Ej. 1&#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\/7427"}],"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=7427"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7427\/revisions"}],"predecessor-version":[{"id":7428,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7427\/revisions\/7428"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/media?parent=7427"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/categories?post=7427"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/tags?post=7427"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}