{"id":1147,"date":"2022-10-26T06:00:52","date_gmt":"2022-10-26T04:00:52","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1147"},"modified":"2022-10-21T11:47:14","modified_gmt":"2022-10-21T09:47:14","slug":"si-r-es-un-anillo-ordenado-entonces-%e2%88%80-a-b-%e2%88%88-r-a-%e2%89%a4-b-%e2%86%92-0-%e2%89%a4-b-a","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/si-r-es-un-anillo-ordenado-entonces-%e2%88%80-a-b-%e2%88%88-r-a-%e2%89%a4-b-%e2%86%92-0-%e2%89%a4-b-a\/","title":{"rendered":"Si R es un anillo ordenado, entonces \u2200 a b \u2208 R, a \u2264 b \u2192 0 \u2264 b &#8211; a"},"content":{"rendered":"<p>Demostrar que si R es un anillo ordenado y a b \u2208 R, entonces<\/p>\n<pre lang=\"text\">\n   a \u2264 b \u2192 0 \u2264 b - a\n<\/pre>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport algebra.order.ring\nvariables {R : Type*} [ordered_ring R]\nvariables a b : R\n\nexample : a \u2264 b \u2192 0 \u2264 b - a :=\nsorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Soluciones con Lean<\/b><\/p>\n<pre lang=\"lean\">\nimport algebra.order.ring\nvariables {R : Type*} [ordered_ring R]\nvariables a b : R\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : a \u2264 b \u2192 0 \u2264 b - a :=\nbegin\n  intro h,\n  calc\n    0   = a - a : (sub_self a).symm\n    ... \u2264 b - a : sub_le_sub_right h a\nend\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : a \u2264 b \u2192 0 \u2264 b - a :=\n-- by library_search\nsub_nonneg.mpr\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : a \u2264 b \u2192 0 \u2264 b - a :=\n-- by hint\nby simp\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\/Si_es_menor_o_igual_entonces_la_diferencia_es_positiva.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. 23.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar que si R es un anillo ordenado y a b \u2208 R, entonces a \u2264 b \u2192 0 \u2264 b &#8211; a Para ello, completar la siguiente teor\u00eda de Lean: import algebra.order.ring variables {R : Type*} [ordered_ring R] variables a b : R example : a \u2264 b \u2192 0 \u2264 b &#8211; a := 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":[294],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1147"}],"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=1147"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1147\/revisions"}],"predecessor-version":[{"id":1151,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1147\/revisions\/1151"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1147"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1147"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1147"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}