{"id":510,"date":"2021-06-30T06:00:18","date_gmt":"2021-06-30T04:00:18","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=510"},"modified":"2021-07-01T16:13:03","modified_gmt":"2021-07-01T14:13:03","slug":"producto-de-potencias-de-la-misma-base-en-monoides","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/producto-de-potencias-de-la-misma-base-en-monoides\/","title":{"rendered":"Producto de potencias de la misma base en monoides"},"content":{"rendered":"<p>En los <a href=\"https:\/\/en.wikipedia.org\/wiki\/Monoid\">monoides<\/a> se define la<br \/>\npotencia con exponentes naturales por recursi\u00f3n:<\/p>\n<pre lang=\"text\">\n   x^0     = 1\n   x^(n+1) = x * x^n\n<\/pre>\n<p>En Lean la potencia x^n se caracteriza por los siguientes lemas:<\/p>\n<pre lang=\"text\">\n   npow_zero' : x ^ 0 = 1\n   npow_succ' : x ^ (succ n) = x * x ^ n\n<\/pre>\n<p>Demostrar que<\/p>\n<pre lang=\"text\">\n    x ^ (m + n) = x ^ m * x ^ n\n<\/pre>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport algebra.group_power.basic\nopen monoid nat\n\nvariables {M : Type} [monoid M]\nvariable  x : M\nvariables (m n : \u2115)\n\nset_option pp.structure_projections false\n\nexample :\n  x ^ (m + n) = x ^ m * x ^ n :=\nsorry\n<\/pre>\n<p>[expand title=\u00bbSoluciones con Lean\u00bb]<\/p>\n<pre lang=\"lean\">\r\nimport algebra.group_power.basic\r\nopen monoid nat\r\n\r\nvariables {M : Type} [monoid M]\r\nvariable  x : M\r\nvariables (m n : \u2115)\r\n\r\n-- Para que no use la notaci\u00f3n con puntos\r\nset_option pp.structure_projections false\r\n\r\n-- 1\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample :\r\n  x ^ (m + n) = x ^ m * x ^ n :=\r\nbegin\r\n  induction m with m HI,\r\n  { calc x ^ (0 + n)\r\n         = x ^ n               : congr_arg ((^) x) (nat.zero_add n)\r\n     ... = 1 * x ^ n           : (monoid.one_mul (x ^ n)).symm\r\n     ... = x ^ 0 * x ^ n       : congr_arg (* (x ^ n)) (monoid.npow_zero' x).symm, },\r\n  { calc x ^ (succ m + n)\r\n         = x ^ succ (m + n)    : congr_arg ((^) x) (succ_add m n)\r\n     ... = x * x ^ (m + n)     : pow_succ x (m + n)\r\n     ... = x * (x ^ m * x ^ n) : congr_arg ((*) x) HI\r\n     ... = (x * x ^ m) * x ^ n : (monoid.mul_assoc x (x ^ m) (x ^ n)).symm\r\n     ... = x ^ succ m * x ^ n  : congr_arg (* x^n) (pow_succ x m).symm, },\r\nend\r\n\r\n-- 2\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample :\r\n  x ^ (m + n) = x ^ m * x ^ n :=\r\nbegin\r\n  induction m with m HI,\r\n  { calc x ^ (0 + n)\r\n         = x ^ n               : by simp only [nat.zero_add]\r\n     ... = 1 * x ^ n           : by simp only [monoid.one_mul]\r\n     ... = x ^ 0 * x ^ n       : by simp [monoid.npow_zero'] },\r\n  { calc x ^ (succ m + n)\r\n         = x ^ succ (m + n)    : by simp only [succ_add]\r\n     ... = x * x ^ (m + n)     : by simp only [pow_succ]\r\n     ... = x * (x ^ m * x ^ n) : by simp only [HI]\r\n     ... = (x * x ^ m) * x ^ n : (monoid.mul_assoc x (x ^ m) (x ^ n)).symm\r\n     ... = x ^ succ m * x ^ n  : by simp only [pow_succ], },\r\nend\r\n\r\n-- 3\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample :\r\n  x ^ (m + n) = x ^ m * x ^ n :=\r\nbegin\r\n  induction m with m HI,\r\n  { calc x ^ (0 + n)\r\n         = x ^ n               : by simp [nat.zero_add]\r\n     ... = 1 * x ^ n           : by simp\r\n     ... = x ^ 0 * x ^ n       : by simp, },\r\n  { calc x ^ (succ m + n)\r\n         = x ^ succ (m + n)    : by simp [succ_add]\r\n     ... = x * x ^ (m + n)     : by simp [pow_succ]\r\n     ... = x * (x ^ m * x ^ n) : by simp [HI]\r\n     ... = (x * x ^ m) * x ^ n : (monoid.mul_assoc x (x ^ m) (x ^ n)).symm\r\n     ... = x ^ succ m * x ^ n  : by simp [pow_succ], },\r\nend\r\n\r\n-- 4\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample :\r\n  x ^ (m + n) = x ^ m * x ^ n :=\r\nbegin\r\n  induction m with m HI,\r\n  { show x ^ (0 + n) = x ^ 0 * x ^ n,\r\n      by simp [nat.zero_add] },\r\n  { show x ^ (succ m + n) = x ^ succ m * x ^ n,\r\n      by finish [succ_add,\r\n                 HI,\r\n                 monoid.mul_assoc,\r\n                 pow_succ], },\r\nend\r\n\r\n-- 5\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample :\r\n  x ^ (m + n) = x ^ m * x ^ n :=\r\npow_add x m n\r\n<\/pre>\n<p>Se puede interactuar con la prueba anterior en <a href=\"https:\/\/www.cs.us.es\/~jalonso\/lean-web-editor\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus\/main\/src\/Producto_de_potencias_de_la_misma_base_en_monoides.lean\" rel=\"noopener noreferrer\" target=\"_blank\">esta sesi\u00f3n con Lean<\/a>,<\/p>\n<p>En los comentarios se pueden escribir otras soluciones, escribiendo el c\u00f3digo entre una l\u00ednea con &#60;pre lang=&quot;isar&quot;&#62; y otra con &#60;\/pre&#62;<br \/>\n[\/expand]<\/p>\n<p>[expand title=\u00bbSoluciones con Isabelle\/HOL\u00bb]<\/p>\n<pre lang=\"isar\">\r\ntheory Producto_de_potencias_de_la_misma_base_en_monoides\r\nimports Main\r\nbegin\r\n\r\ncontext monoid_mult\r\nbegin\r\n\r\n(* 1\u00aa demostraci\u00f3n *)\r\n\r\nlemma \"x ^ (m + n) = x ^ m * x ^ n\"\r\nproof (induct m)\r\n  have \"x ^ (0 + n) = x ^ n\"                 by (simp only: add_0)\r\n  also have \"\u2026 = 1 * x ^ n\"                 by (simp only: mult_1_left)\r\n  also have \"\u2026 = x ^ 0 * x ^ n\"             by (simp only: power_0)\r\n  finally show \"x ^ (0 + n) = x ^ 0 * x ^ n\"\r\n    by this\r\nnext\r\n  fix m\r\n  assume HI : \"x ^ (m + n) = x ^ m * x ^ n\"\r\n  have \"x ^ (Suc m + n) = x ^ Suc (m + n)\"    by (simp only: add_Suc)\r\n  also have \"\u2026 = x *  x ^ (m + n)\"           by (simp only: power_Suc)\r\n  also have \"\u2026 = x *  (x ^ m * x ^ n)\"       by (simp only: HI)\r\n  also have \"\u2026 = (x *  x ^ m) * x ^ n\"       by (simp only: mult_assoc)\r\n  also have \"\u2026 = x ^ Suc m * x ^ n\"          by (simp only: power_Suc)\r\n  finally show \"x ^ (Suc m + n) = x ^ Suc m * x ^ n\"\r\n    by this\r\nqed\r\n\r\n(* 2\u00aa demostraci\u00f3n *)\r\n\r\nlemma \"x ^ (m + n) = x ^ m * x ^ n\"\r\nproof (induct m)\r\n  have \"x ^ (0 + n) = x ^ n\"                  by simp\r\n  also have \"\u2026 = 1 * x ^ n\"                  by simp\r\n  also have \"\u2026 = x ^ 0 * x ^ n\"              by simp\r\n  finally show \"x ^ (0 + n) = x ^ 0 * x ^ n\"\r\n    by this\r\nnext\r\n  fix m\r\n  assume HI : \"x ^ (m + n) = x ^ m * x ^ n\"\r\n  have \"x ^ (Suc m + n) = x ^ Suc (m + n)\"    by simp\r\n  also have \"\u2026 = x *  x ^ (m + n)\"           by simp\r\n  also have \"\u2026 = x *  (x ^ m * x ^ n)\"       using HI by simp\r\n  also have \"\u2026 = (x *  x ^ m) * x ^ n\"       by (simp add: mult_assoc)\r\n  also have \"\u2026 = x ^ Suc m * x ^ n\"          by simp\r\n  finally show \"x ^ (Suc m + n) = x ^ Suc m * x ^ n\"\r\n    by this\r\nqed\r\n\r\n(* 3\u00aa demostraci\u00f3n *)\r\n\r\nlemma \"x ^ (m + n) = x ^ m * x ^ n\"\r\nproof (induct m)\r\n  case 0\r\n  then show ?case\r\n    by simp\r\nnext\r\n  case (Suc m)\r\n  then show ?case\r\n    by (simp add: algebra_simps)\r\nqed\r\n\r\n(* 4\u00aa demostraci\u00f3n *)\r\n\r\nlemma \"x ^ (m + n) = x ^ m * x ^ n\"\r\n  by (induct m) (simp_all add: algebra_simps)\r\n\r\n(* 5\u00aa demostraci\u00f3n *)\r\n\r\nlemma \"x ^ (m + n) = x ^ m * x ^ n\"\r\n  by (simp only: power_add)\r\n\r\nend\r\n\r\nend\r\n<\/pre>\n<p>En los comentarios se pueden escribir otras soluciones, escribiendo el c\u00f3digo entre una l\u00ednea con &#60;pre lang=&quot;isar&quot;&#62; y otra con &#60;\/pre&#62;<br \/>\n[\/expand]<\/p>\n","protected":false},"excerpt":{"rendered":"<p>En los monoides se define la potencia con exponentes naturales por recursi\u00f3n: x^0 = 1 x^(n+1) = x * x^n En Lean la potencia x^n se caracteriza por los siguientes lemas: npow_zero&#8217; : x ^ 0 = 1 npow_succ&#8217; : x ^ (succ n) = x * x ^ n Demostrar que x ^ (m + n) = x ^ m * x ^ n Para ello, completar la siguiente teor\u00eda de Lean: import algebra.group_power.basic open monoid nat variables {M : Type} [monoid M] variable x : M variables (m n : \u2115) set_option pp.structure_projections false example : x ^ (m + n) = x ^ m * x ^ n := sorry [expand title=\u00bbSoluciones con Lean\u00bb] import algebra.group_power.basic open monoid nat variables {M :&#8230;<\/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":[9],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/510"}],"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=510"}],"version-history":[{"count":4,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/510\/revisions"}],"predecessor-version":[{"id":541,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/510\/revisions\/541"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=510"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=510"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=510"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}