{"id":1026,"date":"2022-09-12T06:00:02","date_gmt":"2022-09-12T04:00:02","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1026"},"modified":"2022-09-11T16:24:11","modified_gmt":"2022-09-11T14:24:11","slug":"si-r-es-un-anillo-y-a-b-%e2%88%88-r-entonces-a-b-a-b","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/si-r-es-un-anillo-y-a-b-%e2%88%88-r-entonces-a-b-a-b\/","title":{"rendered":"Si R es un anillo y a, b \u2208 R, entonces a &#8211; b = a + -b"},"content":{"rendered":"<p>Demostrar que si R es un anillo y a, b \u2208 R, entonces<\/p>\n<pre lang=\"text\">\na - b = a + -b\n<\/pre>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport algebra.ring\n\nvariables {R : Type*} [ring R]\nvariables {a b : R}\n\nexample : a - b = a + -b :=\nsorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Soluciones con Lean<\/b><\/p>\n<pre lang=\"lean\">\nimport algebra.ring\n\nvariables {R : Type*} [ring R]\nvariables {a b : R}\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : a - b = a + -b :=\nbegin\n  apply sub_eq_iff_eq_add.mpr,\n  calc a\n       = a + 0        : (add_zero a).symm\n   ... = a + (-b + b) : congr_arg (\u03bb x, a + x) (neg_add_self b).symm\n   ... = a + -b + b   : (add_assoc a (-b) b).symm\nend\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : a - b = a + -b :=\nbegin\n  apply sub_eq_iff_eq_add.mpr,\n  calc a\n       = a + 0        : by rw add_zero\n   ... = a + (-b + b) : by {congr; rw neg_add_self}\n   ... = a + -b + b   : by rw add_assoc\nend\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : a - b = a + -b :=\nbegin\n  rw sub_eq_iff_eq_add,\n  rw add_assoc,\n  rw neg_add_self,\n  rw add_zero,\nend\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : a - b = a + -b :=\nby rw [sub_eq_iff_eq_add, add_assoc, neg_add_self, add_zero]\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample : a - b = a + -b :=\nby simp [sub_eq_iff_eq_add]\n\n-- 6\u00aa demostraci\u00f3n\n-- ===============\n\nexample : a - b = a + -b :=\n-- by library_search\nsub_eq_add_neg a b\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\/Subtraccion_en_anillos.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. 13.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar que si R es un anillo y a, b \u2208 R, entonces a &#8211; b = a + -b Para ello, completar la siguiente teor\u00eda de Lean: import algebra.ring variables {R : Type*} [ring R] variables {a b : R} example : a &#8211; b = a + -b := 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":[284],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1026"}],"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=1026"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1026\/revisions"}],"predecessor-version":[{"id":1094,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1026\/revisions\/1094"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1026"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1026"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1026"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}