s ⊆ f⁻¹[f[s]]
Demostrar que si \(s\) es un subconjunto del dominio de la función \(f\), entonces \(s\) está contenido en la imagen inversa de la imagen de \(s\) por \(f\); es decir,
\[ s ⊆ f⁻¹[f[s]] \]
Para ello, completar la siguiente teoría de Lean4:
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import Mathlib.Data.Set.Function open Set variable {α β : Type _} variable (f : α → β) variable (s : Set α) example : s ⊆ f ⁻¹' (f '' s) := by sorry |
1. Demostración en lenguaje natural
Se demuestra mediante la siguiente cadena de implicaciones
\begin{align}
x ∈ s &⟹ f(x) ∈ f[s] \\
&⟹ x ∈ f⁻¹[f[s]]
\end{align}
2. Demostraciones con Lean4
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import Mathlib.Data.Set.Function open Set variable {α β : Type _} variable (f : α → β) variable (s : Set α) -- 1ª demostración -- =============== example : s ⊆ f ⁻¹' (f '' s) := by intros x xs -- x : α -- xs : x ∈ s -- ⊢ x ∈ f ⁻¹' (f '' s) have h1 : f x ∈ f '' s := mem_image_of_mem f xs show x ∈ f ⁻¹' (f '' s) exact mem_preimage.mp h1 -- 2ª demostración -- =============== example : s ⊆ f ⁻¹' (f '' s) := by intros x xs -- x : α -- xs : x ∈ s -- ⊢ x ∈ f ⁻¹' (f '' s) apply mem_preimage.mpr -- ⊢ f x ∈ f '' s apply mem_image_of_mem -- ⊢ x ∈ s exact xs -- 3ª demostración -- =============== example : s ⊆ f ⁻¹' (f '' s) := by intros x xs -- x : α -- xs : x ∈ s -- ⊢ x ∈ f ⁻¹' (f '' s) apply mem_image_of_mem -- ⊢ x ∈ s exact xs -- 4ª demostración -- =============== example : s ⊆ f ⁻¹' (f '' s) := fun _ ↦ mem_image_of_mem f -- 5ª demostración -- =============== example : s ⊆ f ⁻¹' (f '' s) := by intros x xs -- x : α -- xs : x ∈ s -- ⊢ x ∈ f ⁻¹' (f '' s) show f x ∈ f '' s use x, xs -- 6ª demostración -- =============== example : s ⊆ f ⁻¹' (f '' s) := by intros x xs -- x : α -- xs : x ∈ s -- ⊢ x ∈ f ⁻¹' (f '' s) use x, xs -- 7ª demostración -- =============== example : s ⊆ f ⁻¹' (f '' s) := subset_preimage_image f s -- Lemas usados -- ============ -- variable (x : α) -- variable (t : Set β) -- #check (mem_preimage : x ∈ f ⁻¹' t ↔ f x ∈ t) -- #check (mem_image_of_mem f : x ∈ s → f x ∈ f '' s) -- #check (subset_preimage_image f s : s ⊆ f ⁻¹' (f '' s)) |
Se puede interactuar con las demostraciones anteriores en Lean 4 Web.
3. Demostraciones con Isabelle/HOL
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theory Imagen_inversa_de_la_imagen imports Main begin (* 1ª demostración *) lemma "s ⊆ f -` (f ` s)" proof (rule subsetI) fix x assume "x ∈ s" then have "f x ∈ f ` s" by (simp only: imageI) then show "x ∈ f -` (f ` s)" by (simp only: vimageI) qed (* 2ª demostración *) lemma "s ⊆ f -` (f ` s)" proof fix x assume "x ∈ s" then have "f x ∈ f ` s" by simp then show "x ∈ f -` (f ` s)" by simp qed (* 3ª demostración *) lemma "s ⊆ f -` (f ` s)" by auto end |
