{"id":1822,"date":"2023-12-01T06:00:35","date_gmt":"2023-12-01T04:00:35","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1822"},"modified":"2023-11-21T11:30:37","modified_gmt":"2023-11-21T09:30:37","slug":"01-dic-23","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/01-dic-23\/","title":{"rendered":"\u00ac\u00acP \u2192 P"},"content":{"rendered":"\n<p>Demostrar con Lean4 que &#92;(\u00ac\u00acP \u2192 P&#92;).<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Tactic\nvariable (P : Prop)\n\nexample\n  (h : \u00ac\u00acP)\n  : P :=\nby sorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Demostraci\u00f3n en lenguaje natural<\/b><\/p>\n<p>Por reducci\u00f3n al absurdo. Supongamos &#92;(\u00acP&#92;). Entonces, tenemos una contradicci\u00f3n con la hip\u00f3tesis (&#92;(\u00ac\u00acP&#92;)).<\/p>\n<p><b>Demostraciones con Lean4<\/b><\/p>\n<pre lang=\"lean\">\nimport Mathlib.Tactic\nvariable (P : Prop)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : \u00ac\u00acP)\n  : P :=\nby\n  by_contra h1\n  -- h1 : \u00acP\n  -- \u22a2 False\n  exact (h h1)\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : \u00ac\u00acP)\n  : P :=\nby_contra (fun h1 \u21a6 h h1)\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : \u00ac\u00acP)\n  : P :=\n-- not_not.mp h\nof_not_not h\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : \u00ac\u00acP)\n  : P :=\nby tauto\n\n-- Lemas usados\n-- ============\n\n-- #check (of_not_not : \u00ac\u00acP \u2192 P)\n<\/pre>\n<p><b>Demostraciones interactivas<\/b><\/p>\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\/Eliminacion_doble_negacion.lean\" rel=\"noopener noreferrer\" target=\"_blank\">Lean 4 Web<\/a>.<\/p>\n<p><b>Referencias<\/b><\/p>\n<ul>\n<li> J. Avigad y P. Massot. <a href=\"https:\/\/bit.ly\/3U4UjBk\">Mathematics in Lean<\/a>, p. 33.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar con Lean4 que &#92;(\u00ac\u00acP \u2192 P&#92;). Para ello, completar la siguiente teor\u00eda de Lean4: import Mathlib.Tactic variable (P : Prop) example (h : \u00ac\u00acP) : P := 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\/1822"}],"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=1822"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1822\/revisions"}],"predecessor-version":[{"id":1823,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1822\/revisions\/1823"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1822"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1822"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1822"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}