Si G es un grupo y a ∈ G, entonces a * a⁻¹ = 1
En Lean, se declara que G es un grupo mediante la expresión
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variables {G : Type*} [group G] |
y, como consecuencia, se tiene los siguientes axiomas
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mul_assoc : ∀ a b c : G, a * b * c = a * (b * c) one_mul : ∀ a : G, 1 * a = a mul_left_inv : ∀ a : G, a⁻¹ * a = 1 |
Demostrar que si G es un grupo y a ∈ G, entonces
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a * a⁻¹ = 1 |
Para ello, completar la siguiente teoría de Lean:
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import algebra.group variables {G : Type*} [group G] variables (a b : G) example : a * a⁻¹ = 1 := sorry |
Soluciones con Lean
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import algebra.group variables {G : Type*} [group G] variables (a b : G) -- 1ª demostración -- =============== example : a * a⁻¹ = 1 := calc a * a⁻¹ = 1 * (a * a⁻¹) : (one_mul (a * a⁻¹)).symm ... = (1 * a) * a⁻¹ : (mul_assoc 1 a a⁻¹).symm ... = (((a⁻¹)⁻¹ * a⁻¹) * a) * a⁻¹ : congr_arg (λ x, (x * a) * a⁻¹) (mul_left_inv a⁻¹).symm ... = ((a⁻¹)⁻¹ * (a⁻¹ * a)) * a⁻¹ : congr_fun (congr_arg has_mul.mul (mul_assoc a⁻¹⁻¹ a⁻¹ a)) a⁻¹ ... = ((a⁻¹)⁻¹ * 1) * a⁻¹ : congr_arg (λ x, (a⁻¹⁻¹ * x) * a⁻¹) (mul_left_inv a) ... = (a⁻¹)⁻¹ * (1 * a⁻¹) : mul_assoc (a⁻¹)⁻¹ 1 a⁻¹ ... = (a⁻¹)⁻¹ * a⁻¹ : congr_arg (λ x, (a⁻¹)⁻¹ * x) (one_mul a⁻¹) ... = 1 : mul_left_inv a⁻¹ -- 2ª demostración -- =============== example : a * a⁻¹ = 1 := calc a * a⁻¹ = 1 * (a * a⁻¹) : by rw one_mul ... = (1 * a) * a⁻¹ : by rw mul_assoc ... = (((a⁻¹)⁻¹ * a⁻¹) * a) * a⁻¹ : by rw mul_left_inv ... = ((a⁻¹)⁻¹ * (a⁻¹ * a)) * a⁻¹ : by rw ← mul_assoc ... = ((a⁻¹)⁻¹ * 1) * a⁻¹ : by rw mul_left_inv ... = (a⁻¹)⁻¹ * (1 * a⁻¹) : by rw mul_assoc ... = (a⁻¹)⁻¹ * a⁻¹ : by rw one_mul ... = 1 : by rw mul_left_inv -- 3ª demostración -- =============== example : a * a⁻¹ = 1 := calc a * a⁻¹ = 1 * (a * a⁻¹) : by simp ... = (1 * a) * a⁻¹ : by simp ... = (((a⁻¹)⁻¹ * a⁻¹) * a) * a⁻¹ : by simp ... = ((a⁻¹)⁻¹ * (a⁻¹ * a)) * a⁻¹ : by simp ... = ((a⁻¹)⁻¹ * 1) * a⁻¹ : by simp ... = (a⁻¹)⁻¹ * (1 * a⁻¹) : by simp ... = (a⁻¹)⁻¹ * a⁻¹ : by simp ... = 1 : by simp -- 4ª demostración -- =============== example : a * a⁻¹ = 1 := by simp -- 5ª demostración -- =============== example : a * a⁻¹ = 1 := -- by library_search mul_inv_self a |
Se puede interactuar con la prueba anterior en esta sesión con Lean.
Referencias
- J. Avigad, K. Buzzard, R.Y. Lewis y P. Massot. Mathematics in Lean, p. 14.
