{"id":1973,"date":"2024-01-24T06:00:57","date_gmt":"2024-01-24T04:00:57","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1973"},"modified":"2024-01-19T12:12:28","modified_gmt":"2024-01-19T10:12:28","slug":"24-ene-24","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/24-ene-24\/","title":{"rendered":"(P \u2192 Q) \u2194 \u00acP \u2228 Q"},"content":{"rendered":"\n<p>Demostrar con Lean4 que<br \/>\n\\[ (P \u2192 Q) \u2194 \u00acP \u2228 Q \\]<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\r\nimport Mathlib.Tactic\r\nvariable (P Q : Prop)\r\n\r\nexample\r\n  : (P \u2192 Q) \u2194 \u00acP \u2228 Q :=\r\nby sorry\r\n<\/pre>\n<p><!--more--><\/p>\n<h2>1. Demostraci\u00f3n en lenguaje natural<\/h2>\n<p>Demostraremos cada una de las implicaciones.<\/p>\n<p>(==>) Supongamos que \\(P \u2192 Q\\). Distinguimos dos subcasos seg\u00fan el valor de \\(P\\).<\/p>\n<p>Primer subcaso: suponemos \\(P\\). Entonces, tenemos \\(Q\\) (porque \\(P \u2192 Q\\)) y. por tanto, \\(\u00acP \u2228 Q\\).<\/p>\n<p>Segundo subcaso: suponemos \\(\u00acP\\). Entonces. tenemos \\(\u00acP \u2228 Q\\).<\/p>\n<p>(<==) Supongamos que \\(\u00acP \u2228 Q\\) y \\(P\\) y tenemos que demostrar \\(Q\\). Distinguimos dos subcasos seg\u00fan \\(\u00acP \u2228 Q\\).\n\nPrimer subcaso: Suponemos \\(\u00acP\\). Entonces tenemos una contradicci\u00f3n con \\(P\\).\n\nSegundo subcaso: Suponemos \\(Q\\), que es lo que tenemos que demostrar.\n\n\n\n<h2>2. Demostraciones con Lean4<\/h2>\n<pre lang=\"lean\">\r\nimport Mathlib.Tactic\r\nvariable (P Q : Prop)\r\n\r\n-- 1\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample\r\n  : (P \u2192 Q) \u2194 \u00acP \u2228 Q :=\r\nby\r\n  constructor\r\n  . -- \u22a2 (P \u2192 Q) \u2192 \u00acP \u2228 Q\r\n    intro h1\r\n    -- h1 : P \u2192 Q\r\n    -- \u22a2 \u00acP \u2228 Q\r\n    by_cases h2 : P\r\n    . -- h2 : P\r\n      right\r\n      -- \u22a2 Q\r\n      apply h1\r\n      -- \u22a2 P\r\n      exact h2\r\n    . -- h2 : \u00acP\r\n      left\r\n      -- \u22a2 \u00acP\r\n      exact h2\r\n  . -- \u22a2 \u00acP \u2228 Q \u2192 P \u2192 Q\r\n    intros h3 h4\r\n    -- h3 : \u00acP \u2228 Q\r\n    -- h4 : P\r\n    -- \u22a2 Q\r\n    rcases h3 with h3a | h3b\r\n    . -- h : \u00acP\r\n      exact absurd h4 h3a\r\n    . -- h : Q\r\n      exact h3b\r\n  done\r\n\r\n-- 2\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample\r\n  : (P \u2192 Q) \u2194 \u00acP \u2228 Q :=\r\nby\r\n  constructor\r\n  . -- \u22a2 (P \u2192 Q) \u2192 \u00acP \u2228 Q\r\n    intro h1\r\n    -- h1 : P \u2192 Q\r\n    -- \u22a2 \u00acP \u2228 Q\r\n    by_cases h2: P\r\n    . -- h2 : P\r\n      right\r\n      -- \u22a2 Q\r\n      exact h1 h2\r\n    . -- h2 : \u00acP\r\n      left\r\n      -- \u22a2 \u00acP\r\n      exact h2\r\n  . -- \u22a2 \u00acP \u2228 Q \u2192 P \u2192 Q\r\n    intros h3 h4\r\n    -- h3 : \u00acP \u2228 Q\r\n    -- h4 : P\r\n    -- \u22a2 Q\r\n    cases h3\r\n    . -- h : \u00acP\r\n      contradiction\r\n    . -- h : Q\r\n      assumption\r\n  done\r\n\r\n-- 3\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample\r\n  (P Q : Prop)\r\n  : (P \u2192 Q) \u2194 \u00acP \u2228 Q :=\r\nimp_iff_not_or\r\n\r\n-- 4\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample\r\n  (P Q : Prop)\r\n  : (P \u2192 Q) \u2194 \u00acP \u2228 Q :=\r\nby tauto\r\n<\/pre>\n<h3>Demostraciones interactivas<\/h3>\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\/Implicacion_mediante_disyuncion_y_negacion.lean\" rel=\"noopener noreferrer\" target=\"_blank\">Lean 4 Web<\/a>.<\/p>\n<h3>Referencias<\/h3>\n<ul>\n<li> J. Avigad y P. Massot. <a href=\"https:\/\/bit.ly\/3U4UjBk\">Mathematics in Lean<\/a>, p. 40.<\/li>\n<\/ul>\n<h2>3. Demostraciones con Isabelle\/HOL<\/h2>\n<pre lang=\"isar\">\r\ntheory Implicacion_mediante_disyuncion_y_negacion\r\nimports Main\r\nbegin\r\n\r\n(* 1\u00aa demostraci\u00f3n *)\r\nlemma \"(P \u27f6 Q) \u27f7 (\u00acP \u2228 Q)\"\r\nproof\r\n  assume \"P \u27f6 Q\"\r\n  show \"\u00acP \u2228 Q\"\r\n  proof -\r\n    have \"\u00acP \u2228 P\" by (rule excluded_middle)\r\n    then show \"\u00acP \u2228 Q\"\r\n    proof (rule disjE)\r\n      assume \"\u00acP\"\r\n      then show \"\u00acP \u2228 Q\" by (rule disjI1)\r\n    next\r\n      assume 2: \"P\"\r\n      with `P \u27f6 Q` have \"Q\" by (rule mp)\r\n      then show \"\u00acP \u2228 Q\" by (rule disjI2)\r\n    qed\r\n  qed\r\nnext\r\n  assume \"\u00acP \u2228 Q\"\r\n  show \"P \u27f6 Q\"\r\n  proof\r\n    assume \"P\"\r\n    note `\u00acP \u2228 Q`\r\n    then show \"Q\"\r\n    proof (rule disjE)\r\n      assume \"\u00acP\"\r\n      then show Q using `P` by (rule notE)\r\n    next\r\n      assume \"Q\"\r\n      then show \"Q\" by this\r\n    qed\r\n  qed\r\nqed\r\n\r\n(* 2\u00aa demostraci\u00f3n *)\r\nlemma \"(P \u27f6 Q) \u27f7 (\u00acP \u2228 Q)\"\r\nproof\r\n  assume \"P \u27f6 Q\"\r\n  show \"\u00acP \u2228 Q\"\r\n  proof -\r\n    have \"\u00acP \u2228 P\" by (rule excluded_middle)\r\n    then show \"\u00acP \u2228 Q\"\r\n    proof\r\n      assume \"\u00acP\"\r\n      then show \"\u00acP \u2228 Q\" ..\r\n    next\r\n      assume 2: \"P\"\r\n      with `P \u27f6 Q` have \"Q\" ..\r\n      then show \"\u00acP \u2228 Q\" ..\r\n    qed\r\n  qed\r\nnext\r\n  assume \"\u00acP \u2228 Q\"\r\n  show \"P \u27f6 Q\"\r\n  proof\r\n    assume \"P\"\r\n    note `\u00acP \u2228 Q`\r\n    then show \"Q\"\r\n    proof\r\n      assume \"\u00acP\"\r\n      then show Q using `P` ..\r\n    next\r\n      assume \"Q\"\r\n      then show \"Q\" .\r\n    qed\r\n  qed\r\nqed\r\n\r\n(* 3\u00aa demostraci\u00f3n *)\r\nlemma \"(P \u27f6 Q) \u27f7 (\u00acP \u2228 Q)\"\r\n  by simp\r\n\r\nend\r\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar con Lean4 que \\[ (P \u2192 Q) \u2194 \u00acP \u2228 Q \\] Para ello, completar la siguiente teor\u00eda de Lean4: import Mathlib.Tactic variable (P Q : Prop) example : (P \u2192 Q) \u2194 \u00acP \u2228 Q := by sorry<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","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,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[1],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1973"}],"collection":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/comments?post=1973"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1973\/revisions"}],"predecessor-version":[{"id":1974,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1973\/revisions\/1974"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1973"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1973"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1973"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}