{"id":1020,"date":"2022-09-08T06:00:02","date_gmt":"2022-09-08T04:00:02","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1020"},"modified":"2022-09-11T13:56:13","modified_gmt":"2022-09-11T11:56:13","slug":"si-r-es-un-anillo-entonces-0-0","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/si-r-es-un-anillo-entonces-0-0\/","title":{"rendered":"Si R es un anillo, entonces -0 = 0"},"content":{"rendered":"<p>Demostrar que si R es un anillo, entonces<\/p>\n<pre lang=\"text\">\n-0 = 0\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]\n\nexample : (-0 : R) = 0 :=\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]\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (-0 : R) = 0 :=\nbegin\n  have h : 0 - 0 = (-0 : R) := zero_sub 0,\n  calc (-0 : R)\n       = 0 - 0    : h.symm\n   ... = -(0 - 0) : (neg_sub (0 : R) 0).symm\n   ... = -(-0)    : congr_arg (\u03bb x, -x) h\n   ... = 0        : neg_neg 0\nend\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (-0 : R) = 0 :=\nbegin\n  have h : 0 - 0 = (-0 : R) := by rw zero_sub,\n  calc (-0 : R)\n       = 0 - 0    : by rw h\n   ... = -(0 - 0) : by rw neg_sub\n   ... = -(-0)    : by {congr; rw h}\n   ... = 0        : by rw neg_neg\nend\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (-0 : R) = 0 :=\nby simpa only [zero_sub, neg_neg] using (neg_sub (0 : R) 0).symm\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (-0 : R) = 0 :=\nneg_zero\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (-0 : R) = 0 :=\nby simp\n\n-- 6\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (-0 : R) = 0 :=\nbegin\n  apply neg_eq_of_add_eq_zero_right,\n  rw add_zero,\nend\n\n-- 7\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (-0 : R) = 0 :=\nneg_eq_of_add_eq_zero_right (add_zero 0)\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\/Opuesto_de_cero.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. 12.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar que si R es un anillo, entonces -0 = 0 Para ello, completar la siguiente teor\u00eda de Lean: import algebra.ring variables {R : Type*} [ring R] example : (-0 : R) = 0 := 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\/1020"}],"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=1020"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1020\/revisions"}],"predecessor-version":[{"id":1084,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1020\/revisions\/1084"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1020"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1020"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1020"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}