{"id":1887,"date":"2023-12-28T06:00:02","date_gmt":"2023-12-28T04:00:02","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1887"},"modified":"2023-12-26T19:27:18","modified_gmt":"2023-12-26T17:27:18","slug":"28-dic-23","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/28-dic-23\/","title":{"rendered":"Si |x + 3| < 5, entonces -8 < x < 2"},"content":{"rendered":"\n<p>Demostrar con Lean4 que si &#92;(|x + 3| &lt; 5&#92;), entonces &#92;(-8 &lt; x &lt; 2&#92;).<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Real.Basic\nvariable (x y : \u211d)\n\nexample\n  : |x + 3| < 5 \u2192 -8 < x \u2227 x < 2 :=\nby sorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Demostraci\u00f3n en lenguaje natural<\/b><\/p>\n<p>Supongamos que<br \/>\n&#92;[ |x + 3| &lt; 5 &#92;]<br \/>\nentonces<br \/>\n&#92;[ -5 &lt; x + 3 &lt; 5 &#92;]<br \/>\npor tanto<br \/>\n&#92;[ -8 &lt; x &lt; 2 &#92;]<\/p>\n<p><b>Demostraciones con Lean4<\/b><\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Real.Basic\nvariable (x y : \u211d)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  : |x + 3| < 5 \u2192 -8 < x \u2227 x < 2 :=\nby\n  rw [abs_lt]\n  -- \u22a2 -5 < x + 3 \u2227 x + 3 < 5 \u2192 -8 < x \u2227 x < 2\n  intro h\n  -- h : -5 < x + 3 \u2227 x + 3 < 5\n  -- \u22a2 -8 < x \u2227 x < 2\n  constructor\n  . -- \u22a2 -8 < x\n    linarith\n  . -- x < 2\n    linarith\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  : |x + 3| < 5 \u2192 -8 < x \u2227 x < 2 :=\nby\n  rw [abs_lt]\n  intro h\n  constructor <;> linarith\n\n-- Comentario: La composici\u00f3n (constructor <;> linarith) aplica constructor y a\n-- continuaci\u00f3n le aplica linarith a cada subojetivo.\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  : |x + 3| < 5 \u2192 -8 < x \u2227 x < 2 :=\nby\n  rw [abs_lt]\n  exact fun _ \u21a6 \u27e8by linarith, by linarith\u27e9\n\n-- Lemas usados\n-- ============\n\n-- #check (abs_lt: |x| < y \u2194 -y < x \u2227 x < y)\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\/Acotacion_del_valor_absoluto.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. 37.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar con Lean4 que si &#92;(|x + 3| &lt; 5&#92;), entonces &#92;(-8 &lt; x &lt; 2&#92;). Para ello, completar la siguiente teor\u00eda de Lean4: import Mathlib.Data.Real.Basic variable (x y : \u211d) example : |x + 3| < 5 \u2192 -8 < x \u2227 x < 2 := by sorry\n<\/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\/1887"}],"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=1887"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1887\/revisions"}],"predecessor-version":[{"id":1889,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1887\/revisions\/1889"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1887"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1887"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1887"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}