{"id":988,"date":"2022-09-01T06:00:25","date_gmt":"2022-09-01T04:00:25","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=988"},"modified":"2022-09-11T13:58:07","modified_gmt":"2022-09-11T11:58:07","slug":"si-r-es-un-anillo-y-a-b-c-%e2%88%88-r-tales-que-a-b-c-b-entonces-a-c","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/si-r-es-un-anillo-y-a-b-c-%e2%88%88-r-tales-que-a-b-c-b-entonces-a-c\/","title":{"rendered":"Si R es un anillo y a, b, c \u2208 R tales que a + b = c + b, entonces a = c"},"content":{"rendered":"<p>Demostrar que si R es un anillo y a, b, c \u2208 R tales que<\/p>\n<pre lang=\"text\">\n   a + b = c + b\n<\/pre>\n<p>entonces<\/p>\n<pre lang=\"text\">\n   a = c\n<\/pre>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport algebra.ring\nimport tactic\n\nvariables {R : Type*} [ring R]\nvariables {a b c : R}\n\nexample\n  (h : a + b = c + b)\n  : a = c :=\nsorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Soluciones con Lean<\/b><\/p>\n<pre lang=\"lean\">\nimport algebra.ring\nimport tactic\n\nvariables {R : Type*} [ring R]\nvariables {a b c : R}\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : a + b = c + b)\n  : a = c :=\ncalc a\n     = a + 0        : by rw add_zero\n ... = a + (b + -b) : by rw add_right_neg\n ... = (a + b) + -b : by rw add_assoc\n ... = (c + b) + -b : by rw h\n ... = c + (b + -b) : by rw \u2190 add_assoc\n ... = c + 0        : by rw \u2190 add_right_neg\n ... = c            : by rw add_zero\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : a + b = c + b)\n  : a = c :=\ncalc a\n     = a + 0        : by simp\n ... = a + (b + -b) : by simp\n ... = (a + b) + -b : by simp\n ... = (c + b) + -b : by rw h\n ... = c + (b + -b) : by simp\n ... = c + 0        : by simp\n ... = c            : by simp\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nlemma aux : (a + b) + -b = a :=\nby finish\n\nexample\n  (h : a + b = c + b)\n  : a = c :=\ncalc a\n     = (a + b) + -b : aux.symm\n ... = (c + b) + -b : congr_arg (\u03bb x, x + -b) h\n ... = c            : aux\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : a + b = c + b)\n  : a = c :=\nby finish\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\/Cancelativa_de_la_suma_por_la_derecha.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 y a, b, c \u2208 R tales que a + b = c + b entonces a = c Para ello, completar la siguiente teor\u00eda de Lean: import algebra.ring import tactic variables {R : Type*} [ring R] variables {a b c : R} example (h : a + b = c + b) : a = 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":[284],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/988"}],"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=988"}],"version-history":[{"count":5,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/988\/revisions"}],"predecessor-version":[{"id":1089,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/988\/revisions\/1089"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=988"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=988"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=988"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}