Square root on rational numbers

This file defines the square root function on rational numbers Rat.sqrt and proves several theorems about it.

def Rat.sqrt (q : ℚ) : ℚ

Square root function on rational numbers, defined by taking the (integer) square root of the numerator and the square root (on natural numbers) of the denominator.

theorem Rat.sqrt_eq (q : ℚ) : Rat.sqrt (q * q) = |q|

theorem Rat.exists_mul_self (x : ℚ) : (∃ q, q * q = x) ↔ Rat.sqrt x * Rat.sqrt x = x

theorem Rat.sqrt_nonneg (q : ℚ) : 0 ≤ Rat.sqrt q