Square root on rational numbers

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

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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.

Equations

• Rat.sqrt q = mkRat (Int.sqrt q.num) (Nat.sqrt q.den)

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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 = x) ↔ Rat.sqrt x * Rat.sqrt x = x

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