{"id":2298,"date":"2024-03-04T06:00:35","date_gmt":"2024-03-04T04:00:35","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=2298"},"modified":"2024-03-02T18:21:27","modified_gmt":"2024-03-02T16:21:27","slug":"04-mar-24","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/04-mar-24\/","title":{"rendered":"(s \\ t) \u222a (t \\ s) = (s \u222a t) \\ (s \u2229 t)"},"content":{"rendered":"\n

Demostrar con Lean4 que
\n\[ (s \setminus t) \u222a (t \setminus s) = (s \u222a t) \setminus (s \u2229 t) \]<\/p>\n

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

\nimport Mathlib.Data.Set.Basic\nopen Set\n\nvariable {\u03b1 : Type}\nvariable (s t : Set \u03b1)\n\nexample : (s \\\\ t) \u222a (t \\\\ s) = (s \u222a t) \\\\ (s \u2229 t) :=\nby sorry\n<\/pre>\n
<\/p>\n
1. Demostraci\u00f3n en lenguaje natural<\/h2>\n
Tenemos que demostrar que, para todo \(x\),
\n\[ x \u2208 (s \setminus t) \u222a (t \setminus s) \u2194 x \u2208 (s \u222a t) \setminus (s \u2229 t) \]
\nSe demuestra mediante la siguiente cadena de equivalencias:
\n\begin{align}
\n     &x \u2208 (s \setminus t) \u222a (t \setminus s) \\
\n   \u2194 &x \u2208 (s \setminus t) \u2228 x \u2208 (t \setminus s) \\
\n   \u2194 &(x \u2208 s \u2227 x \u2209 t) \u2228 x \u2208 (t \setminus s) \\
\n   \u2194 &(x \u2208 s \u2228 x \u2208 (t \ s)) \u2227 (x \u2209 t \u2228 x \u2208 (t \setminus s)) \\
\n   \u2194 &(x \u2208 s \u2228 (x \u2208 t \u2227 x \u2209 s)) \u2227 (x \u2209 t \u2228 (x \u2208 t \u2227 x \u2209 s)) \\
\n   \u2194 &((x \u2208 s \u2228 x \u2208 t) \u2227 (x \u2208 s \u2228 x \u2209 s)) \u2227 ((x \u2209 t \u2228 x \u2208 t) \u2227 (x \u2209 t \u2228 x \u2209 s)) \\
\n   \u2194 &(x \u2208 s \u2228 x \u2208 t) \u2227 (x \u2209 t \u2228 x \u2209 s) \\
\n   \u2194 &(x \u2208 s \u222a t) \u2227 (x \u2209 t \u2228 x \u2209 s) \\
\n   \u2194 &(x \u2208 s \u222a t) \u2227 (x \u2209 s \u2228 x \u2209 t) \\
\n   \u2194 &(x \u2208 s \u222a t) \u2227 \u00ac(x \u2208 s \u2227 x \u2208 t) \\
\n   \u2194 &(x \u2208 s \u222a t) \u2227 \u00ac(x \u2208 s \u2229 t) \\
\n   \u2194 &x \u2208 (s \u222a t) \setminus (s \u2229 t)
\n\end{align}<\/p>\n
2. Demostraciones con Lean4<\/h2>\n
\nimport Mathlib.Data.Set.Basic\nopen Set\n\nvariable {\u03b1 : Type}\nvariable (s t : Set \u03b1)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (s \\ t) \u222a (t \\ s) = (s \u222a t) \\ (s \u2229 t) :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 (s \\ t) \u222a (t \\ s) \u2194 x \u2208 (s \u222a t) \\ (s \u2229 t)\n  calc x \u2208 (s \\ t) \u222a (t \\ s)\n     \u2194 x \u2208 (s \\ t) \u2228 x \u2208 (t \\ s) :=\n         by exact mem_union x (s \\ t) (t \\ s)\n   _ \u2194 (x \u2208 s \u2227 x \u2209 t) \u2228 x \u2208 (t \\ s) :=\n         by simp only [mem_diff]\n   _ \u2194 (x \u2208 s \u2228 x \u2208 (t \\ s)) \u2227 (x \u2209 t \u2228 x \u2208 (t \\ s)) :=\n         by exact and_or_right\n   _ \u2194 (x \u2208 s \u2228 (x \u2208 t \u2227 x \u2209 s)) \u2227 (x \u2209 t \u2228 (x \u2208 t \u2227 x \u2209 s)) :=\n         by simp only [mem_diff]\n   _ \u2194 ((x \u2208 s \u2228 x \u2208 t) \u2227 (x \u2208 s \u2228 x \u2209 s)) \u2227\n       ((x \u2209 t \u2228 x \u2208 t) \u2227 (x \u2209 t \u2228 x \u2209 s)) :=\n         by simp_all only [or_and_left]\n   _ \u2194 ((x \u2208 s \u2228 x \u2208 t) \u2227 True) \u2227\n       (True \u2227 (x \u2209 t \u2228 x \u2209 s)) :=\n         by simp only [em (x \u2208 s), em' (x \u2208 t)]\n   _ \u2194 (x \u2208 s \u2228 x \u2208 t) \u2227 (x \u2209 t \u2228 x \u2209 s) :=\n         by simp only [and_true_iff (x \u2208 s \u2228 x \u2208 t),\n                       true_and_iff (\u00acx \u2208 t \u2228 \u00acx \u2208 s)]\n   _ \u2194 (x \u2208 s \u222a t) \u2227 (x \u2209 t \u2228 x \u2209 s) :=\n         by simp only [mem_union]\n   _ \u2194 (x \u2208 s \u222a t) \u2227 (x \u2209 s \u2228 x \u2209 t) :=\n         by simp only [or_comm]\n   _ \u2194 (x \u2208 s \u222a t) \u2227 \u00ac(x \u2208 s \u2227 x \u2208 t) :=\n         by simp only [not_and_or]\n   _ \u2194 (x \u2208 s \u222a t) \u2227 \u00ac(x \u2208 s \u2229 t) :=\n         by simp only [mem_inter_iff]\n   _ \u2194 x \u2208 (s \u222a t) \\ (s \u2229 t)     :=\n         by simp only [mem_diff]\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (s \\ t) \u222a (t \\ s) = (s \u222a t) \\ (s \u2229 t) :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 (s \\ t) \u222a (t \\ s) \u2194 x \u2208 (s \u222a t) \\ (s \u2229 t)\n  constructor\n  . -- \u22a2 x \u2208 (s \\ t) \u222a (t \\ s) \u2192 x \u2208 (s \u222a t) \\ (s \u2229 t)\n    rintro (\u27e8xs, xnt\u27e9 | \u27e8xt, xns\u27e9)\n    . -- xs : x \u2208 s\n      -- xnt : \u00acx \u2208 t\n      -- \u22a2 x \u2208 (s \u222a t) \\ (s \u2229 t)\n      constructor\n      . -- \u22a2 x \u2208 s \u222a t\n        left\n        -- \u22a2 x \u2208 s\n        exact xs\n      . -- \u22a2 \u00acx \u2208 s \u2229 t\n        rintro \u27e8-, xt\u27e9\n        -- xt : x \u2208 t\n        -- \u22a2 False\n        exact xnt xt\n    . -- xt : x \u2208 t\n      -- xns : \u00acx \u2208 s\n      -- \u22a2 x \u2208 (s \u222a t) \\ (s \u2229 t)\n      constructor\n      . -- \u22a2 x \u2208 s \u222a t\n        right\n        -- \u22a2 x \u2208 t\n        exact xt\n      . -- \u22a2 \u00acx \u2208 s \u2229 t\n        rintro \u27e8xs, -\u27e9\n        -- xs : x \u2208 s\n        -- \u22a2 False\n        exact xns xs\n  . -- \u22a2 x \u2208 (s \u222a t) \\ (s \u2229 t) \u2192 x \u2208 (s \\ t) \u222a (t \\ s)\n    rintro \u27e8xs | xt, nxst\u27e9\n    . -- xs : x \u2208 s\n      -- \u22a2 x \u2208 (s \\ t) \u222a (t \\ s)\n      left\n      -- \u22a2 x \u2208 s \\ t\n      use xs\n      -- \u22a2 \u00acx \u2208 t\n      intro xt\n      -- xt : x \u2208 t\n      -- \u22a2 False\n      apply nxst\n      -- \u22a2 x \u2208 s \u2229 t\n      constructor\n      . -- \u22a2 x \u2208 s\n        exact xs\n      . -- \u22a2 x \u2208 t\n        exact xt\n    . -- nxst : \u00acx \u2208 s \u2229 t\n      -- xt : x \u2208 t\n      -- \u22a2 x \u2208 (s \\ t) \u222a (t \\ s)\n      right\n      -- \u22a2 x \u2208 t \\ s\n      use xt\n      -- \u22a2 \u00acx \u2208 s\n      intro xs\n      -- xs : x \u2208 s\n      -- \u22a2 False\n      apply nxst\n      -- \u22a2 x \u2208 s \u2229 t\n      constructor\n      . -- \u22a2 x \u2208 s\n        exact xs\n      . -- \u22a2 x \u2208 t\n        exact xt\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (s \\ t) \u222a (t \\ s) = (s \u222a t) \\ (s \u2229 t) :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 (s \\ t) \u222a (t \\ s) \u2194 x \u2208 (s \u222a t) \\ (s \u2229 t)\n  constructor\n  . -- \u22a2 x \u2208 (s \\ t) \u222a (t \\ s) \u2192 x \u2208 (s \u222a t) \\ (s \u2229 t)\n    rintro (\u27e8xs, xnt\u27e9 | \u27e8xt, xns\u27e9)\n    . -- xt : x \u2208 t\n      -- xns : \u00acx \u2208 s\n      -- \u22a2 x \u2208 (s \u222a t) \\ (s \u2229 t)\n      aesop\n    . -- xt : x \u2208 t\n      -- xns : \u00acx \u2208 s\n      -- \u22a2 x \u2208 (s \u222a t) \\ (s \u2229 t)\n      aesop\n  . rintro \u27e8xs | xt, nxst\u27e9\n    . -- xs : x \u2208 s\n      -- \u22a2 x \u2208 (s \\ t) \u222a (t \\ s)\n      aesop\n    . -- nxst : \u00acx \u2208 s \u2229 t\n      -- xt : x \u2208 t\n      -- \u22a2 x \u2208 (s \\ t) \u222a (t \\ s)\n      aesop\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (s \\ t) \u222a (t \\ s) = (s \u222a t) \\ (s \u2229 t) :=\nby\n  ext x\n  -- x : \u03b1\n  -- \u22a2 x \u2208 (s \\ t) \u222a (t \\ s) \u2194 x \u2208 (s \u222a t) \\ (s \u2229 t)\n  constructor\n  . -- \u22a2 x \u2208 (s \\ t) \u222a (t \\ s) \u2192 x \u2208 (s \u222a t) \\ (s \u2229 t)\n    rintro (\u27e8xs, xnt\u27e9 | \u27e8xt, xns\u27e9) <;> aesop\n  . -- \u22a2 x \u2208 (s \u222a t) \\ (s \u2229 t) \u2192 x \u2208 (s \\ t) \u222a (t \\ s)\n    rintro \u27e8xs | xt, nxst\u27e9 <;> aesop\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (s \\ t) \u222a (t \\ s) = (s \u222a t) \\ (s \u2229 t) :=\nby\n  ext\n  constructor\n  . aesop\n  . aesop\n\n-- 6\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (s \\ t) \u222a (t \\ s) = (s \u222a t) \\ (s \u2229 t) :=\nby\n  ext\n  constructor <;> aesop\n\n-- 7\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (s \\ t) \u222a (t \\ s) = (s \u222a t) \\ (s \u2229 t) :=\nby\n  rw [ext_iff]\n  -- \u22a2 \u2200 (x : \u03b1), x \u2208 (s \\ t) \u222a (t \\ s) \u2194 x \u2208 (s \u222a t) \\ (s \u2229 t)\n  intro\n  -- x : \u03b1\n  -- \u22a2 x \u2208 (s \\ t) \u222a (t \\ s) \u2194 x \u2208 (s \u222a t) \\ (s \u2229 t)\n  rw [iff_def]\n  -- \u22a2 (x \u2208 (s \\ t) \u222a (t \\ s) \u2192 x \u2208 (s \u222a t) \\ (s \u2229 t)) \u2227\n  --   (x \u2208 (s \u222a t) \\ (s \u2229 t) \u2192 x \u2208 (s \\ t) \u222a (t \\ s))\n  aesop\n\n-- Lemas usados\n-- ============\n\n-- variable (x : \u03b1)\n-- variable (a b c : Prop)\n-- #check (mem_union x s t : x \u2208 s \u222a t \u2194 x \u2208 s \u2228 x \u2208 t)\n-- #check (mem_diff x : x \u2208 s \\ t \u2194 x \u2208 s \u2227 \u00acx \u2208 t)\n-- #check (and_or_right : (a \u2227 b) \u2228 c \u2194 (a \u2228 c) \u2227 (b \u2228 c))\n-- #check (or_and_left : a \u2228 (b \u2227 c) \u2194 (a \u2228 b) \u2227 (a \u2228 c))\n-- #check (em a : a \u2228 \u00ac a)\n-- #check (em' a : \u00ac a \u2228 a)\n-- #check (and_true_iff a : a \u2227 True \u2194 a)\n-- #check (true_and_iff a : True \u2227 a \u2194 a)\n-- #check (or_comm : a \u2228 b \u2194 b \u2228 a)\n-- #check (not_and_or : \u00ac(a \u2227 b) \u2194 \u00aca \u2228 \u00acb)\n-- #check (mem_inter_iff x s t : x \u2208 s \u2229 t \u2194 x \u2208 s \u2227 x \u2208 t)\n-- #check (ext_iff : s = t \u2194 \u2200 (x : \u03b1), x \u2208 s \u2194 x \u2208 t)\n-- #check (iff_def : (a \u2194 b) \u2194 (a \u2192 b) \u2227 (b \u2192 a))\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 Diferencia_de_union_e_interseccion\nimports Main\nbegin\n\n(* 1 demostraci\u00f3n *)\nlemma \"(s - t) \u222a (t - s) = (s \u222a t) - (s \u2229 t)\"\nproof (rule equalityI)\n  show \"(s - t) \u222a (t - s) \u2286 (s \u222a t) - (s \u2229 t)\"\n  proof (rule subsetI)\n    fix x\n    assume \"x \u2208 (s - t) \u222a (t - s)\"\n    then show \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n    proof (rule UnE)\n      assume \"x \u2208 s - t\"\n      then show \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n      proof (rule DiffE)\n        assume \"x \u2208 s\"\n        assume \"x \u2209 t\"\n        have \"x \u2208 s \u222a t\"\n          using \u2039x \u2208 s\u203a by (simp only: UnI1)\n        moreover\n        have \"x \u2209 s \u2229 t\"\n        proof (rule notI)\n          assume \"x \u2208 s \u2229 t\"\n          then have \"x \u2208 t\"\n            by (simp only: IntD2)\n          with \u2039x \u2209 t\u203a show False\n            by (rule notE)\n        qed\n        ultimately show \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n          by (rule DiffI)\n      qed\n    next\n      assume \"x \u2208 t - s\"\n      then show \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n      proof (rule DiffE)\n        assume \"x \u2208 t\"\n        assume \"x \u2209 s\"\n        have \"x \u2208 s \u222a t\"\n          using \u2039x \u2208 t\u203a by (simp only: UnI2)\n        moreover\n        have \"x \u2209 s \u2229 t\"\n        proof (rule notI)\n          assume \"x \u2208 s \u2229 t\"\n          then have \"x \u2208 s\"\n            by (simp only: IntD1)\n          with \u2039x \u2209 s\u203a show False\n            by (rule notE)\n        qed\n        ultimately show \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n          by (rule DiffI)\n      qed\n    qed\n  qed\nnext\n  show \"(s \u222a t) - (s \u2229 t) \u2286 (s - t) \u222a (t - s)\"\n  proof (rule subsetI)\n    fix x\n    assume \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n    then show \"x \u2208 (s - t) \u222a (t - s)\"\n    proof (rule DiffE)\n      assume \"x \u2208 s \u222a t\"\n      assume \"x \u2209 s \u2229 t\"\n      note \u2039x \u2208 s \u222a t\u203a\n      then show \"x \u2208 (s - t) \u222a (t - s)\"\n      proof (rule UnE)\n        assume \"x \u2208 s\"\n        have \"x \u2209 t\"\n        proof (rule notI)\n          assume \"x \u2208 t\"\n          with \u2039x \u2208 s\u203a have \"x \u2208 s \u2229 t\"\n            by (rule IntI)\n          with \u2039x \u2209 s \u2229 t\u203a show False\n            by (rule notE)\n        qed\n        with \u2039x \u2208 s\u203a have \"x \u2208 s - t\"\n          by (rule DiffI)\n        then show \"x \u2208 (s - t) \u222a (t - s)\"\n          by (simp only: UnI1)\n      next\n        assume \"x \u2208 t\"\n        have \"x \u2209 s\"\n        proof (rule notI)\n          assume \"x \u2208 s\"\n          then have \"x \u2208 s \u2229 t\"\n            using \u2039x \u2208 t\u203a by (rule IntI)\n          with \u2039x \u2209 s \u2229 t\u203a show False\n            by (rule notE)\n        qed\n        with \u2039x \u2208 t\u203a have \"x \u2208 t - s\"\n          by (rule DiffI)\n        then show \"x \u2208 (s - t) \u222a (t - s)\"\n          by (rule UnI2)\n      qed\n    qed\n  qed\nqed\n\n(* 2 demostraci\u00f3n *)\nlemma \"(s - t) \u222a (t - s) = (s \u222a t) - (s \u2229 t)\"\nproof\n  show \"(s - t) \u222a (t - s) \u2286 (s \u222a t) - (s \u2229 t)\"\n  proof\n    fix x\n    assume \"x \u2208 (s - t) \u222a (t - s)\"\n    then show \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n    proof\n      assume \"x \u2208 s - t\"\n      then show \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n      proof\n        assume \"x \u2208 s\"\n        assume \"x \u2209 t\"\n        have \"x \u2208 s \u222a t\"\n          using \u2039x \u2208 s\u203a by simp\n        moreover\n        have \"x \u2209 s \u2229 t\"\n        proof\n          assume \"x \u2208 s \u2229 t\"\n          then have \"x \u2208 t\"\n            by simp\n          with \u2039x \u2209 t\u203a show False\n            by simp\n        qed\n        ultimately show \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n          by simp\n      qed\n    next\n      assume \"x \u2208 t - s\"\n      then show \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n      proof\n        assume \"x \u2208 t\"\n        assume \"x \u2209 s\"\n        have \"x \u2208 s \u222a t\"\n          using \u2039x \u2208 t\u203a by simp\n        moreover\n        have \"x \u2209 s \u2229 t\"\n        proof\n          assume \"x \u2208 s \u2229 t\"\n          then have \"x \u2208 s\"\n            by simp\n          with \u2039x \u2209 s\u203a show False\n            by simp\n        qed\n        ultimately show \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n          by simp\n      qed\n    qed\n  qed\nnext\n  show \"(s \u222a t) - (s \u2229 t) \u2286 (s - t) \u222a (t - s)\"\n  proof\n    fix x\n    assume \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n    then show \"x \u2208 (s - t) \u222a (t - s)\"\n    proof\n      assume \"x \u2208 s \u222a t\"\n      assume \"x \u2209 s \u2229 t\"\n      note \u2039x \u2208 s \u222a t\u203a\n      then show \"x \u2208 (s - t) \u222a (t - s)\"\n      proof\n        assume \"x \u2208 s\"\n        have \"x \u2209 t\"\n        proof\n          assume \"x \u2208 t\"\n          with \u2039x \u2208 s\u203a have \"x \u2208 s \u2229 t\"\n            by simp\n          with \u2039x \u2209 s \u2229 t\u203a show False\n            by simp\n        qed\n        with \u2039x \u2208 s\u203a have \"x \u2208 s - t\"\n          by simp\n        then show \"x \u2208 (s - t) \u222a (t - s)\"\n          by simp\n      next\n        assume \"x \u2208 t\"\n        have \"x \u2209 s\"\n        proof\n          assume \"x \u2208 s\"\n          then have \"x \u2208 s \u2229 t\"\n            using \u2039x \u2208 t\u203a by simp\n          with \u2039x \u2209 s \u2229 t\u203a show False\n            by simp\n        qed\n        with \u2039x \u2208 t\u203a have \"x \u2208 t - s\"\n          by simp\n        then show \"x \u2208 (s - t) \u222a (t - s)\"\n          by simp\n      qed\n    qed\n  qed\nqed\n\n(* 3\u00aa demostraci\u00f3n *)\nlemma \"(s - t) \u222a (t - s) = (s \u222a t) - (s \u2229 t)\"\nproof\n  show \"(s - t) \u222a (t - s) \u2286 (s \u222a t) - (s \u2229 t)\"\n  proof\n    fix x\n    assume \"x \u2208 (s - t) \u222a (t - s)\"\n    then show \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n    proof\n      assume \"x \u2208 s - t\"\n      then show \"x \u2208 (s \u222a t) - (s \u2229 t)\" by simp\n    next\n      assume \"x \u2208 t - s\"\n      then show \"x \u2208 (s \u222a t) - (s \u2229 t)\" by simp\n    qed\n  qed\nnext\n  show \"(s \u222a t) - (s \u2229 t) \u2286 (s - t) \u222a (t - s)\"\n  proof\n    fix x\n    assume \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n    then show \"x \u2208 (s - t) \u222a (t - s)\"\n    proof\n      assume \"x \u2208 s \u222a t\"\n      assume \"x \u2209 s \u2229 t\"\n      note \u2039x \u2208 s \u222a t\u203a\n      then show \"x \u2208 (s - t) \u222a (t - s)\"\n      proof\n        assume \"x \u2208 s\"\n        then show \"x \u2208 (s - t) \u222a (t - s)\"\n          using \u2039x \u2209 s \u2229 t\u203a by simp\n      next\n        assume \"x \u2208 t\"\n        then show \"x \u2208 (s - t) \u222a (t - s)\"\n          using \u2039x \u2209 s \u2229 t\u203a by simp\n      qed\n    qed\n  qed\nqed\n\n(* 4\u00aa demostraci\u00f3n *)\n\nlemma \"(s - t) \u222a (t - s) = (s \u222a t) - (s \u2229 t)\"\nproof\n  show \"(s - t) \u222a (t - s) \u2286 (s \u222a t) - (s \u2229 t)\"\n  proof\n    fix x\n    assume \"x \u2208 (s - t) \u222a (t - s)\"\n    then show \"x \u2208 (s \u222a t) - (s \u2229 t)\" by auto\n  qed\nnext\n  show \"(s \u222a t) - (s \u2229 t) \u2286 (s - t) \u222a (t - s)\"\n  proof\n    fix x\n    assume \"x \u2208 (s \u222a t) - (s \u2229 t)\"\n    then show \"x \u2208 (s - t) \u222a (t - s)\" by auto\n  qed\nqed\n\n(* 5\u00aa demostraci\u00f3n *)\n\nlemma \"(s - t) \u222a (t - s) = (s \u222a t) - (s \u2229 t)\"\nproof\n  show \"(s - t) \u222a (t - s) \u2286 (s \u222a t) - (s \u2229 t)\" by auto\nnext\n  show \"(s \u222a t) - (s \u2229 t) \u2286 (s - t) \u222a (t - s)\" by auto\nqed\n\n(* 6\u00aa demostraci\u00f3n *)\n\nlemma \"(s - t) \u222a (t - s) = (s \u222a t) - (s \u2229 t)\"\n  by auto\n\nend\n<\/pre>\n","protected":false},"excerpt":{"rendered":"
Demostrar con Lean4 que \[ (s \setminus t) \u222a (t \setminus s) = (s \u222a t) \setminus (s \u2229 t) \] Para ello, completar la siguiente teor\u00eda de Lean4: import Mathlib.Data.Set.Basic open Set variable {\u03b1 : Type} variable (s t : Set \u03b1) example : (s \\\\ t) \u222a (t \\\\ s) = (s \u222a t) \\\\ (s \u2229 t) := by 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":"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":[7],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/2298"}],"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=2298"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/2298\/revisions"}],"predecessor-version":[{"id":2300,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/2298\/revisions\/2300"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=2298"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=2298"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=2298"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}