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Conmutatividad_de_la_interseccion.thy
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Conmutatividad_de_la_interseccion.thy
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(* Conmutatividad_de_la_interseccion.thy
Conmutatividad de la intersección.
José A. Alonso Jiménez
Sevilla, 24 de mayo de 2021
---------------------------------------------------------------------
*)
text \<open>-- ---------------------------------------------------------------
-- Demostrar que
-- s \<inter> t = t \<inter> s
-- --------------------------------------------------------------------\<close>
theory Conmutatividad_de_la_interseccion
imports Main
begin
(* 1\<ordfeminine> demostración *)
lemma "s \<inter> t = t \<inter> s"
proof (rule set_eqI)
fix x
show "x \<in> s \<inter> t \<longleftrightarrow> x \<in> t \<inter> s"
proof (rule iffI)
assume h : "x \<in> s \<inter> t"
then have xs : "x \<in> s"
by (simp only: IntD1)
have xt : "x \<in> t"
using h by (simp only: IntD2)
then show "x \<in> t \<inter> s"
using xs by (rule IntI)
next
assume h : "x \<in> t \<inter> s"
then have xt : "x \<in> t"
by (simp only: IntD1)
have xs : "x \<in> s"
using h by (simp only: IntD2)
then show "x \<in> s \<inter> t"
using xt by (rule IntI)
qed
qed
(* 2\<ordfeminine> demostración *)
lemma "s \<inter> t = t \<inter> s"
proof (rule set_eqI)
fix x
show "x \<in> s \<inter> t \<longleftrightarrow> x \<in> t \<inter> s"
proof
assume h : "x \<in> s \<inter> t"
then have xs : "x \<in> s"
by simp
have xt : "x \<in> t"
using h by simp
then show "x \<in> t \<inter> s"
using xs by simp
next
assume h : "x \<in> t \<inter> s"
then have xt : "x \<in> t"
by simp
have xs : "x \<in> s"
using h by simp
then show "x \<in> s \<inter> t"
using xt by simp
qed
qed
(* 3\<ordfeminine> demostración *)
lemma "s \<inter> t = t \<inter> s"
proof (rule equalityI)
show "s \<inter> t \<subseteq> t \<inter> s"
proof (rule subsetI)
fix x
assume h : "x \<in> s \<inter> t"
then have xs : "x \<in> s"
by (simp only: IntD1)
have xt : "x \<in> t"
using h by (simp only: IntD2)
then show "x \<in> t \<inter> s"
using xs by (rule IntI)
qed
next
show "t \<inter> s \<subseteq> s \<inter> t"
proof (rule subsetI)
fix x
assume h : "x \<in> t \<inter> s"
then have xt : "x \<in> t"
by (simp only: IntD1)
have xs : "x \<in> s"
using h by (simp only: IntD2)
then show "x \<in> s \<inter> t"
using xt by (rule IntI)
qed
qed
(* 4\<ordfeminine> demostración *)
lemma "s \<inter> t = t \<inter> s"
proof
show "s \<inter> t \<subseteq> t \<inter> s"
proof
fix x
assume h : "x \<in> s \<inter> t"
then have xs : "x \<in> s"
by simp
have xt : "x \<in> t"
using h by simp
then show "x \<in> t \<inter> s"
using xs by simp
qed
next
show "t \<inter> s \<subseteq> s \<inter> t"
proof
fix x
assume h : "x \<in> t \<inter> s"
then have xt : "x \<in> t"
by simp
have xs : "x \<in> s"
using h by simp
then show "x \<in> s \<inter> t"
using xt by simp
qed
qed
(* 5\<ordfeminine> demostración *)
lemma "s \<inter> t = t \<inter> s"
proof
show "s \<inter> t \<subseteq> t \<inter> s"
proof
fix x
assume "x \<in> s \<inter> t"
then show "x \<in> t \<inter> s"
by simp
qed
next
show "t \<inter> s \<subseteq> s \<inter> t"
proof
fix x
assume "x \<in> t \<inter> s"
then show "x \<in> s \<inter> t"
by simp
qed
qed
(* 6\<ordfeminine> demostración *)
lemma "s \<inter> t = t \<inter> s"
by (fact Int_commute)
(* 7\<ordfeminine> demostración *)
lemma "s \<inter> t = t \<inter> s"
by (fact inf_commute)
(* 8\<ordfeminine> demostración *)
lemma "s \<inter> t = t \<inter> s"
by auto
end