Propiedad semidistributiva de la intersección sobre la unión
Demostrar que s ∩ (t ∪ u) ⊆ (s ∩ t) ∪ (s ∩ u)
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Para ello, completar la siguiente teoría de Lean:
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import data.set.basic open set variable {α : Type} variables s t u : set α example : s ∩ (t ∪ u) ⊆ (s ∩ t) ∪ (s ∩ u) := sorry |
1. Soluciones con Lean
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import data.set.basic import tactic open set variable {α : Type} variables s t u : set α -- 1ª demostración -- =============== example : s ∩ (t ∪ u) ⊆ (s ∩ t) ∪ (s ∩ u) := begin intros x hx, cases hx with hxs hxtu, cases hxtu with hxt hxu, { left, split, { exact hxs, }, { exact hxt, }}, { right, split, { exact hxs, }, { exact hxu, }}, end -- 2ª demostración -- =============== example : s ∩ (t ∪ u) ⊆ (s ∩ t) ∪ (s ∩ u) := begin rintros x ⟨hxs, hxt | hxu⟩, { left, exact ⟨hxs, hxt⟩, }, { right, exact ⟨hxs, hxu⟩, }, end -- 3ª demostración -- =============== example : s ∩ (t ∪ u) ⊆ (s ∩ t) ∪ (s ∩ u) := begin rintros x ⟨hxs, hxt | hxu⟩, { exact or.inl ⟨hxs, hxt⟩, }, { exact or.inr ⟨hxs, hxu⟩, }, end -- 4ª demostración -- =============== example : s ∩ (t ∪ u) ⊆ (s ∩ t) ∪ (s ∩ u) := begin intros x hx, by finish, end -- 5ª demostración -- =============== example : s ∩ (t ∪ u) ⊆ (s ∩ t) ∪ (s ∩ u) := by rw inter_union_distrib_left |
El código de las demostraciones se encuentra en GitHub y puede ejecutarse con el Lean Web editor.
La construcción de las demostraciones se muestra en el siguiente vídeo
2. Soluciones con Isabelle/HOL
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theory Propiedad_semidistributiva_de_la_interseccion_sobre_la_union imports Main begin (* 1ª demostración *) lemma "s ∩ (t ∪ u) ⊆ (s ∩ t) ∪ (s ∩ u)" proof (rule subsetI) fix x assume hx : "x ∈ s ∩ (t ∪ u)" then have xs : "x ∈ s" by (simp only: IntD1) have xtu: "x ∈ t ∪ u" using hx by (simp only: IntD2) then have "x ∈ t ∨ x ∈ u" by (simp only: Un_iff) then show " x ∈ s ∩ t ∪ s ∩ u" proof (rule disjE) assume xt : "x ∈ t" have xst : "x ∈ s ∩ t" using xs xt by (simp only: Int_iff) then show "x ∈ (s ∩ t) ∪ (s ∩ u)" by (simp only: UnI1) next assume xu : "x ∈ u" have xst : "x ∈ s ∩ u" using xs xu by (simp only: Int_iff) then show "x ∈ (s ∩ t) ∪ (s ∩ u)" by (simp only: UnI2) qed qed (* 2ª demostración *) lemma "s ∩ (t ∪ u) ⊆ (s ∩ t) ∪ (s ∩ u)" proof fix x assume hx : "x ∈ s ∩ (t ∪ u)" then have xs : "x ∈ s" by simp have xtu: "x ∈ t ∪ u" using hx by simp then have "x ∈ t ∨ x ∈ u" by simp then show " x ∈ s ∩ t ∪ s ∩ u" proof assume xt : "x ∈ t" have xst : "x ∈ s ∩ t" using xs xt by simp then show "x ∈ (s ∩ t) ∪ (s ∩ u)" by simp next assume xu : "x ∈ u" have xst : "x ∈ s ∩ u" using xs xu by simp then show "x ∈ (s ∩ t) ∪ (s ∩ u)" by simp qed qed (* 3ª demostración *) lemma "s ∩ (t ∪ u) ⊆ (s ∩ t) ∪ (s ∩ u)" proof (rule subsetI) fix x assume hx : "x ∈ s ∩ (t ∪ u)" then have xs : "x ∈ s" by (simp only: IntD1) have xtu: "x ∈ t ∪ u" using hx by (simp only: IntD2) then show " x ∈ s ∩ t ∪ s ∩ u" proof (rule UnE) assume xt : "x ∈ t" have xst : "x ∈ s ∩ t" using xs xt by (simp only: Int_iff) then show "x ∈ (s ∩ t) ∪ (s ∩ u)" by (simp only: UnI1) next assume xu : "x ∈ u" have xst : "x ∈ s ∩ u" using xs xu by (simp only: Int_iff) then show "x ∈ (s ∩ t) ∪ (s ∩ u)" by (simp only: UnI2) qed qed (* 4ª demostración *) lemma "s ∩ (t ∪ u) ⊆ (s ∩ t) ∪ (s ∩ u)" proof fix x assume hx : "x ∈ s ∩ (t ∪ u)" then have xs : "x ∈ s" by simp have xtu: "x ∈ t ∪ u" using hx by simp then show " x ∈ s ∩ t ∪ s ∩ u" proof (rule UnE) assume xt : "x ∈ t" have xst : "x ∈ s ∩ t" using xs xt by simp then show "x ∈ (s ∩ t) ∪ (s ∩ u)" by simp next assume xu : "x ∈ u" have xst : "x ∈ s ∩ u" using xs xu by simp then show "x ∈ (s ∩ t) ∪ (s ∩ u)" by simp qed qed (* 5ª demostración *) lemma "s ∩ (t ∪ u) ⊆ (s ∩ t) ∪ (s ∩ u)" by (simp only: Int_Un_distrib) (* 6ª demostración *) lemma "s ∩ (t ∪ u) ⊆ (s ∩ t) ∪ (s ∩ u)" by auto end |