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) :

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.

Equations

rat.sqrt q = int.sqrt q.num /. ↑(nat.sqrt q.denom)

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theorem rat.sqrt_eq (q : ℚ) :

rat.sqrt (q * q) = abs q

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theorem rat.exists_mul_self (x : ℚ) :

(∃ (q : ℚ), q * q = x) ↔ (rat.sqrt x) * rat.sqrt x = x

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theorem rat.sqrt_nonneg (q : ℚ) :

0 ≤ rat.sqrt q