abs_le_sqrt — Mathlib

English

ℤ is provided with a linear order, i.e., any two integers are comparable under ≤, and the order is total and antisymmetric.

Русский

ℤ оснащены линейным порядком: любые два целых числа сравнимы по ≤, порядок тотален и антисимметричен.

LaTeX

$$$\forall a,b \in \mathbb{Z}, a \le b \lor b \le a.$$$

Lean4

theorem abs_le_sqrt {a b : ℤ} (hn : 0 ≤ b) : |a| ≤ b.sqrt ↔ a * a ≤ b := by
  rw [← abs_mul_abs_self, eq_natCast_toNat.mpr hn, eq_natCast_toNat.mpr (abs_nonneg a), Int.sqrt_natCast, ←
    Int.natCast_mul, Nat.cast_le, Nat.cast_le, Nat.le_sqrt]

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