{"id":2520,"date":"2024-05-30T06:00:42","date_gmt":"2024-05-30T04:00:42","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=2520"},"modified":"2024-05-29T09:33:36","modified_gmt":"2024-05-29T07:33:36","slug":"30-may-24","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/30-may-24\/","title":{"rendered":"Si x, y \u2208 \u211d tales que (\u2200 z)[y < z \u2192 x \u2264 z], entonces x \u2264 y"},"content":{"rendered":"\n

Demostrar con Lean4 que si \(x, y \u2208 \u211d\) tales que \((\u2200 z)[y < z \u2192 x \u2264 z]\), entonces \(x \u2264 y\).<\/p>\n

Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n

\nimport Mathlib.Data.Real.Basic\n\nvariable {x y : \u211d}\n\nexample\n  (h : \u2200 z, y < z \u2192 x \u2264 z) :\n  x \u2264 y :=\nby sorry\n<\/pre>\n
<\/p>\n
1. Demostraci\u00f3n en lenguaje natural<\/h2>\n
Lo demostraremos por reducci\u00f3n al absurdo. Para ello, supongamos que
\n\[ x \u2270 y \]
\nEntonces
\n\[ y < x \]
\ny, por la densidad de \(\u211d\), existe un \(a\) tal que
\n\[ y < a < x \]
\nPuesto que \(y < a\), por la hip\u00f3tesis, se tiene que
\n\[ x \u2264 a \]
\nen contradicci\u00f3n con
\n\[ a < x \]<\/p>\n
2. Demostraciones con Lean4<\/h2>\n
\nimport Mathlib.Data.Real.Basic\n\nvariable {x y : \u211d}\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : \u2200 z, y < z \u2192 x \u2264 z) :\n  x \u2264 y :=\nby\n  by_contra h1\n  -- h1 : \u00acx \u2264 y\n  -- \u22a2 False\n  have hxy : x > y := not_le.mp h1\n  -- \u22a2 \u00acx > y\n  cases' (exists_between hxy) with a ha\n  -- a : \u211d\n  -- ha : y < a \u2227 a < x\n  apply (lt_irrefl a)\n  -- \u22a2 a < a\n  calc a\n       < x := ha.2\n     _ \u2264 a := h a ha.1\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : \u2200 z, y < z \u2192 x \u2264 z) :\n  x \u2264 y :=\nby\n  apply le_of_not_gt\n  -- \u22a2 \u00acx > y\n  intro hxy\n  -- hxy : x > y\n  -- \u22a2 False\n  cases' (exists_between hxy) with a ha\n  -- a : \u211d\n  -- ha : y < a \u2227 a < x\n  apply (lt_irrefl a)\n  -- \u22a2 a < a\n  calc a\n       < x := ha.2\n     _ \u2264 a := h a ha.1\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : \u2200 z, y < z \u2192 x \u2264 z) :\n  x \u2264 y :=\nby\n  apply le_of_not_gt\n  -- \u22a2 \u00acx > y\n  intro hxy\n  -- hxy : x > y\n  -- \u22a2 False\n  cases' (exists_between hxy) with a ha\n  -- ha : y < a \u2227 a < x\n  apply (lt_irrefl a)\n  -- \u22a2 a < a\n  exact lt_of_lt_of_le ha.2 (h a ha.1)\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : \u2200 z, y < z \u2192 x \u2264 z) :\n  x \u2264 y :=\nby\n  apply le_of_not_gt\n  -- \u22a2 \u00acx > y\n  intro hxy\n  -- hxy : x > y\n  -- \u22a2 False\n  cases' (exists_between hxy) with a ha\n  -- a : \u211d\n  -- ha : y < a \u2227 a < x\n  exact (lt_irrefl a) (lt_of_lt_of_le ha.2 (h a ha.1))\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : \u2200 z, y < z \u2192 x \u2264 z) :\n  x \u2264 y :=\nby\n  apply le_of_not_gt\n  -- \u22a2 \u00acx > y\n  intro hxy\n  -- hxy : x > y\n  -- \u22a2 False\n  rcases (exists_between hxy) with \u27e8a, hya, hax\u27e9\n  -- a : \u211d\n  -- hya : y < a\n  -- hax : a < x\n  exact (lt_irrefl a) (lt_of_lt_of_le hax (h a hya))\n\n-- 6\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : \u2200 z, y < z \u2192 x \u2264 z) :\n  x \u2264 y :=\nle_of_forall_le_of_dense h\n\n-- Lemas usados\n-- ============\n\n-- variable (z : \u211d)\n-- #check (exists_between : x < y \u2192 \u2203 z, x < z \u2227 z < y)\n-- #check (le_of_forall_le_of_dense : (\u2200 z, y < z \u2192 x \u2264 z) \u2192 x \u2264 y)\n-- #check (le_of_not_gt : \u00acx > y \u2192 x \u2264 y)\n-- #check (lt_irrefl x : \u00acx < x)\n-- #check (lt_of_lt_of_le : x < y \u2192 y \u2264 z \u2192 x < z)\n-- #check (not_le : \u00acx \u2264 y \u2194 y < x)\n<\/pre>\n
Se puede interactuar con las demostraciones anteriores en Lean 4 Web<\/a>.<\/p>\n
3. Demostraciones con Isabelle\/HOL<\/h2>\n
\ntheory Propiedad_de_la_densidad_de_los_reales\nimports Main HOL.Real\nbegin\n\n(* 1\u00aa demostraci\u00f3n *)\n\nlemma\n  fixes x y :: real\n  assumes \"\u2200 z. y < z \u27f6 x \u2264 z\"\n  shows \"x \u2264 y\"\nproof (rule linorder_class.leI; intro notI)\n  assume \"y < x\"\n  then have \"\u2203z. y < z \u2227 z < x\"\n    by (rule dense)\n  then obtain a where ha : \"y < a \u2227 a < x\"\n    by (rule exE)\n  have \"\u00ac a < a\"\n    by (rule order.irrefl)\n  moreover\n  have \"a < a\"\n  proof -\n    have \"y < a \u27f6 x \u2264 a\"\n      using assms by (rule allE)\n    moreover\n    have \"y < a\"\n      using ha by (rule conjunct1)\n    ultimately have \"x \u2264 a\"\n      by (rule mp)\n    moreover\n    have \"a < x\"\n      using ha by (rule conjunct2)\n    ultimately show \"a < a\"\n      by (simp only: less_le_trans)\n  qed\n  ultimately show False\n    by (rule notE)\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\n\nlemma\n  fixes x y :: real\n  assumes \"\u22c0 z. y < z \u27f9 x \u2264 z\"\n  shows \"x \u2264 y\"\nproof (rule linorder_class.leI; intro notI)\n  assume \"y < x\"\n  then have \"\u2203z. y < z \u2227 z < x\"\n    by (rule dense)\n  then obtain a where hya : \"y < a\" and hax : \"a < x\"\n    by auto\n  have \"\u00ac a < a\"\n    by (rule order.irrefl)\n  moreover\n  have \"a < a\"\n  proof -\n    have \"a < x\"\n      using hax .\n    also have \"\u2026 \u2264 a\"\n      using assms[OF hya] .\n    finally show \"a < a\" .\n  qed\n  ultimately show False\n    by (rule notE)\nqed\n\n(* 3\u00aa demostraci\u00f3n *)\n\nlemma\n  fixes x y :: real\n  assumes \"\u22c0 z. y < z \u27f9 x \u2264 z\"\n  shows \"x \u2264 y\"\nproof (rule linorder_class.leI; intro notI)\n  assume \"y < x\"\n  then have \"\u2203z. y < z \u2227 z < x\"\n    by (rule dense)\n  then obtain a where hya : \"y < a\" and hax : \"a < x\"\n    by auto\n  have \"\u00ac a < a\"\n    by (rule order.irrefl)\n  moreover\n  have \"a < a\"\n    using hax assms[OF hya] by (rule less_le_trans)\n  ultimately show False\n    by (rule notE)\nqed\n\n(* 4\u00aa demostraci\u00f3n *)\n\nlemma\n  fixes x y :: real\n  assumes \"\u22c0 z. y < z \u27f9 x \u2264 z\"\n  shows \"x \u2264 y\"\nby (meson assms dense not_less)\n\n(* 5\u00aa demostraci\u00f3n *)\n\nlemma\n  fixes x y :: real\n  assumes \"\u22c0 z. y < z \u27f9 x \u2264 z\"\n  shows \"x \u2264 y\"\nusing assms by (rule dense_ge)\n\n(* 6\u00aa demostraci\u00f3n *)\n\nlemma\n  fixes x y :: real\n  assumes \"\u2200 z. y < z \u27f6 x \u2264 z\"\n  shows \"x \u2264 y\"\nusing assms by (simp only: dense_ge)\n\nend\n<\/pre>\n","protected":false},"excerpt":{"rendered":"
Demostrar con Lean4 que si \(x, y \u2208 \u211d\) tales que \((\u2200 z)[y < z \u2192 x \u2264 z]\), entonces \(x \u2264 y\). Para ello, completar la siguiente teor\u00eda de Lean4: import Mathlib.Data.Real.Basic variable {x y : \u211d} example (h : \u2200 z, y < z \u2192 x \u2264 z) : x \u2264 y := by sorry\n<\/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":"default","_kad_post_title":"default","_kad_post_layout":"default","_kad_post_sidebar_id":"","_kad_post_content_style":"default","_kad_post_vertical_padding":"default","_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":[24],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/2520"}],"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=2520"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/2520\/revisions"}],"predecessor-version":[{"id":2521,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/2520\/revisions\/2521"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=2520"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=2520"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=2520"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}