En los anillos ordenados, {a ≤ b, 0 ≤ c} ⊢ ac ≤ bc
Demostrar con Lean4 que, en los anillos ordenados,
\[ \{a ≤ b, 0 ≤ c\} ⊢ ac ≤ bc \]
Para ello, completar la siguiente teoría de Lean4:
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import Mathlib.Algebra.Order.Ring.Defs variable {R : Type _} [StrictOrderedRing R] variable (a b c : R) example (h1 : a ≤ b) (h2 : 0 ≤ c) : a * c ≤ b * c := by sorry |
Demostración en lenguaje natural
Se usarán los siguientes lemas:
\begin{align}
&0 ≤ a – b ↔ b ≤ a \tag{L1} \\
&0 ≤ a → 0 ≤ b → 0 ≤ ab \tag{L2} \\
&(a – b)c = ac – bc \tag{L3}
\end{align}
Supongamos que
\begin{align}
a &≤ b \tag{1} \\
0 &≤ c
\end{align}
De (1), por L1, se tiene
\[ 0 ≤ b – a \]
y con (2), por L2, se tiene
\[ 0 ≤ (b – a)c \]
que, por L3, da
\[ 0 ≤ bc – ac \]
y, aplicándole L1, se tiene
\[ ac ≤ bc \]
Demostraciones con Lean4
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import Mathlib.Algebra.Order.Ring.Defs variable {R : Type _} [StrictOrderedRing R] variable (a b c : R) -- 1ª demostración -- =============== example (h1 : a ≤ b) (h2 : 0 ≤ c) : a * c ≤ b * c := by have h3 : 0 ≤ b - a := sub_nonneg.mpr h1 have h4 : 0 ≤ b * c - a * c := calc 0 ≤ (b - a) * c := mul_nonneg h3 h2 _ = b * c - a * c := sub_mul b a c show a * c ≤ b * c exact sub_nonneg.mp h4 -- 2ª demostración -- =============== example (h1 : a ≤ b) (h2 : 0 ≤ c) : a * c ≤ b * c := by have h3 : 0 ≤ b - a := sub_nonneg.mpr h1 have h4 : 0 ≤ (b - a) * c := mul_nonneg h3 h2 -- h4 : 0 ≤ b * c - a * c rw [sub_mul] at h4 -- a * c ≤ b * c exact sub_nonneg.mp h4 -- 3ª demostración -- =============== example (h1 : a ≤ b) (h2 : 0 ≤ c) : a * c ≤ b * c := by -- 0 ≤ b * c - a * c apply sub_nonneg.mp -- 0 ≤ (b - a) * c rw [← sub_mul] apply mul_nonneg . -- 0 ≤ b - a exact sub_nonneg.mpr h1 . -- 0 ≤ c exact h2 -- 4ª demostración -- =============== example (h1 : a ≤ b) (h2 : 0 ≤ c) : a * c ≤ b * c := by apply sub_nonneg.mp rw [← sub_mul] apply mul_nonneg (sub_nonneg.mpr h1) h2 -- 5ª demostración example (h1 : a ≤ b) (h2 : 0 ≤ c) : a * c ≤ b * c := -- by apply? mul_le_mul_of_nonneg_right h1 h2 -- Lemas usados -- ============ -- #check (mul_le_mul_of_nonneg_right : a ≤ b → 0 ≤ c → a * c ≤ b * c) -- #check (mul_nonneg : 0 ≤ a → 0 ≤ b → 0 ≤ a * b) -- #check (sub_mul a b c : (a - b) * c = a * c - b * c) -- #check (sub_nonneg : 0 ≤ a - b ↔ b ≤ a) |
Demostraciones interactivas
Se puede interactuar con las demostraciones anteriores en Lean 4 Web.
Referencias
- J. Avigad y P. Massot. Mathematics in Lean, p. 22.
