{"id":1115,"date":"2022-10-04T06:00:25","date_gmt":"2022-10-04T04:00:25","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1115"},"modified":"2022-09-25T17:06:43","modified_gmt":"2022-09-25T15:06:43","slug":"si-a-b-c-%e2%88%88-%e2%84%9d-entonces-min-a-b-c-min-a-c-b-c","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/si-a-b-c-%e2%88%88-%e2%84%9d-entonces-min-a-b-c-min-a-c-b-c\/","title":{"rendered":"Si a, b, c \u2208 \u211d, entonces min a b + c = min (a + c) (b + c)"},"content":{"rendered":"<p>Sean a, b y c n\u00fameros reales. Demostrar que<\/p>\n<pre lang=\"text\">\n   min a b + c = min (a + c) (b + c)\n<\/pre>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport data.real.basic\n\nvariables a b c : \u211d\n\nexample :\n  min a b + c = min (a + c) (b + c) :=\nsorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Soluciones con Lean<\/b><\/p>\n<pre lang=\"lean\">\nimport data.real.basic\n\nvariables a b c : \u211d\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nlemma aux1a :\n  min a b + c \u2264 min (a + c) (b + c) :=\nbegin\n  have h1 : min a b  \u2264 a :=\n    min_le_left a b,\n  have h2 : min a b + c \u2264 a + c :=\n    add_le_add_right h1 c,\n  have h3 : min a b  \u2264 b :=\n    min_le_right a b,\n  have h4 : min a b + c \u2264 b + c :=\n    add_le_add_right h3 c,\n  show min a b + c \u2264 min (a + c) (b + c),\n    by exact le_min h2 h4,\nend\n\nlemma aux2a :\n  min (a + c) (b + c) \u2264 min a b + c :=\nbegin\n  have h1 : min (a + c) (b + c) + -c \u2264 min a b,\n  { calc min (a + c) (b + c) + -c\n         \u2264 min (a + c + -c) (b + c + -c) : aux1a (a + c) (b + c) (-c)\n     ... = min a b                       : by ring_nf },\n  show min (a + c) (b + c) \u2264 min a b + c,\n    by exact add_neg_le_iff_le_add.mp h1,\nend\n\nexample :\n  min a b + c = min (a + c) (b + c) :=\nbegin\n  apply le_antisymm,\n  { exact aux1a a b c, },\n  { exact aux2a a b c, },\nend\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nlemma aux1b :\n  min a b + c \u2264 min (a + c) (b + c) :=\nbegin\n  apply le_min,\n  { apply add_le_add_right,\n    exact min_le_left a b, },\n  { apply add_le_add_right,\n    exact min_le_right a b, },\nend\n\nlemma aux2b :\n  min (a + c) (b + c) \u2264 min a b + c :=\nbegin\n  have h1 : min (a + c) (b + c) + -c \u2264 min a b,\n  { calc min (a + c) (b + c) + -c\n         \u2264 min (a + c + -c) (b + c + -c) : aux1b (a + c) (b + c) (-c)\n     ... = min a b                       : by ring_nf },\n  exact add_neg_le_iff_le_add.mp h1,\nend\n\nexample :\n  min a b + c = min (a + c) (b + c) :=\nbegin\n  apply le_antisymm,\n  { exact aux1b a b c, },\n  { exact aux2b a b c, },\nend\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nlemma aux1c :\n  min a b + c \u2264 min (a + c) (b + c) :=\nle_min (add_le_add_right (min_le_left a b) c)\n       (add_le_add_right (min_le_right a b) c)\n\nlemma aux2c :\n  min (a + c) (b + c) \u2264 min a b + c :=\nbegin\n  have h1 : min (a + c) (b + c) + -c \u2264 min a b,\n  { calc min (a + c) (b + c) + -c\n         \u2264 min (a + c + -c) (b + c + -c) : aux1c (a + c) (b + c) (-c)\n     ... = min a b                       : by ring_nf },\n  exact add_neg_le_iff_le_add.mp h1,\nend\n\nexample :\n  min a b + c = min (a + c) (b + c) :=\nle_antisymm (aux1b a b c) (aux2b a b c)\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : min a b + c = min (a + c) (b + c) :=\nbegin\n  by_cases (a \u2264 b),\n  { have h1 : a + c \u2264 b + c,\n      apply add_le_add_right h,\n    calc min a b + c = a + c               : by simp [min_eq_left h]\n                 ... = min (a + c) (b + c) : by simp [min_eq_left h1]},\n  { have h2: b \u2264 a,\n      linarith,\n    have h3 : b + c \u2264 a + c,\n      { exact add_le_add_right h2 c },\n    calc min a b + c = b + c               : by simp [min_eq_right h2]\n                 ... = min (a + c) (b + c) : by simp [min_eq_right h3]},\nend\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample : min a b + c = min (a + c) (b + c) :=\n(min_add_add_right a b c).symm\n<\/pre>\n<p>Se puede interactuar con la prueba anterior en <a href=\"https:\/\/leanprover-community.github.io\/lean-web-editor\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus\/main\/src\/Minimo_de_suma.lean\" rel=\"noopener noreferrer\" target=\"_blank\">esta sesi\u00f3n con Lean<\/a>.<\/p>\n<p><b>Referencias<\/b><\/p>\n<ul>\n<li>J. Avigad, K. Buzzard, R.Y. Lewis y P. Massot. <a href=\"https:\/\/bit.ly\/3U4UjBk\">Mathematics in Lean<\/a>, p. 20.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Sean a, b y c n\u00fameros reales. Demostrar que min a b + c = min (a + c) (b + c) Para ello, completar la siguiente teor\u00eda de Lean: import data.real.basic variables a b c : \u211d example : min a b + c = min (a + c) (b + c) := 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":[286],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1115"}],"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=1115"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1115\/revisions"}],"predecessor-version":[{"id":1116,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1115\/revisions\/1116"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1115"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1115"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1115"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}