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Si R es un anillo y a ∈ R, entonces a.0 = 0

Demostrar con Lean4 que si R es un anillo y a ∈ R, entonces \[ a·0 = 0 \]

Para ello, completar la siguiente teoría de Lean4:

import Mathlib.Algebra.Ring.Defs
import Mathlib.Tactic

variable {R : Type _} [Ring R]
variable (a : R)

example : a * 0 = 0 :=
sorry

Demostración en lenguaje natural

Basta aplicar la propiedad cancelativa a \[a.0 + a.0 = a.0 + 0\] que se demuestra mediante la siguiente cadena de igualdades \begin{align} a.0 + a.0 &= a.(0 + 0) &&\text{[por la distributiva]} \\ &= a.0 &&\text{[por suma con cero]} \\ &= a.0 + 0 &&\text{[por suma con cero]} \end{align}

Demostraciones con Lean4

import Mathlib.Algebra.Ring.Defs
import Mathlib.Tactic

variable {R : Type _} [Ring R]
variable (a : R)

-- 1ª demostración
-- ===============

example : a * 0 = 0 :=
by
  have h : a * 0 + a * 0 = a * 0 + 0 :=
    calc a * 0 + a * 0 = a * (0 + 0) := by rw [mul_add a 0 0]
                     _ = a * 0       := by rw [add_zero 0]
                     _ = a * 0 + 0   := by rw [add_zero (a * 0)]
  rw [add_left_cancel h]

-- 2ª demostración
-- ===============

example : a * 0 = 0 :=
by
  have h : a * 0 + a * 0 = a * 0 + 0 :=
    calc a * 0 + a * 0 = a * (0 + 0) := by rw [← mul_add]
                     _ = a * 0       := by rw [add_zero]
                     _ = a * 0 + 0   := by rw [add_zero]
  rw [add_left_cancel h]

-- 3ª demostración
-- ===============

example : a * 0 = 0 :=
by
  have h : a * 0 + a * 0 = a * 0 + 0 :=
    by rw [← mul_add, add_zero, add_zero]
  rw [add_left_cancel h]

-- 4ª demostración
-- ===============

example : a * 0 = 0 :=
by
  have : a * 0 + a * 0 = a * 0 + 0 :=
    calc a * 0 + a * 0 = a * (0 + 0) := by simp
                     _ = a * 0       := by simp
                     _ = a * 0 + 0   := by simp
  simp

-- 5ª demostración
-- ===============

example : a * 0 = 0 :=
  mul_zero a

-- 6ª demostración
-- ===============

example : a * 0 = 0 :=
by simp

Demostraciones interactivas

Se puede interactuar con las demostraciones anteriores en Lean 4 Web.

Referencias