53 lines
2.1 KiB
Text
53 lines
2.1 KiB
Text
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/-
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Copyright (c) 2015 Jeremy Avigad. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Jeremy Avigad
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A proof that the square root of an integer is irrational, unless the integer is a perfect square.
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-/
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import data.rat
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open nat eq.ops
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/- First, a textbook proof that sqrt 2 is irrational. -/
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theorem sqrt_two_irrational_aux {a b : ℕ} (co : coprime a b) : a * a ≠ 2 * (b * b) :=
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assume H : a * a = 2 * (b * b),
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have even (a * a), from even_of_exists (exists.intro _ H),
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have even a, from even_of_even_mul_self this,
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obtain c (aeq : a = 2 * c), from exists_of_even this,
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have 2 * (2 * (c * c)) = 2 * (b * b), by rewrite [-H, aeq, mul.assoc, mul.left_comm c],
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have 2 * (c * c) = b * b, from eq_of_mul_eq_mul_left dec_trivial this,
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have even (b * b), from even_of_exists (exists.intro _ (eq.symm this)),
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have even b, from even_of_even_mul_self this,
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have 2 ∣ gcd a b, from dvd_gcd (dvd_of_even `even a`) (dvd_of_even `even b`),
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have 2 ∣ 1, from co ▸ this,
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absurd `2 ∣ 1` dec_trivial
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/- Let's state this in terms of rational numbers. The problem is that we now have to mediate between
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rat, int, and nat. -/
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section
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open rat int
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theorem sqrt_two_irrational (q : ℚ): q^2 ≠ 2 :=
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suppose q^2 = 2,
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let a := num q, b := denom q in
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have b ≠ 0, from ne_of_gt (denom_pos q),
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assert bnz : b ≠ (0 : ℚ), from assume H, `b ≠ 0` (of_int.inj H),
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have b * b ≠ (0 : ℚ), from rat.mul_ne_zero bnz bnz,
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have (a * a) / (b * b) = 2,
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by rewrite [*of_int_mul, -div_mul_div bnz bnz, -eq_num_div_denom, -this, rat.pow_two],
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have a * a = 2 * (b * b), from eq.symm (mul_eq_of_eq_div `b * b ≠ (0 : ℚ)` this⁻¹),
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assert a * a = 2 * (b * b), from of_int.inj this, -- now in the integers
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let a' := nat_abs a, b' := nat_abs b in
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have H : a' * a' = 2 * (b' * b'),
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begin
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apply of_nat.inj,
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rewrite [-+nat_abs_mul, int.of_nat_mul, +of_nat_nat_abs, +int.abs_mul_self],
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exact this,
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end,
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have coprime a b, from !coprime_num_denom,
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have nat.coprime a' b', from of_nat.inj this,
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show false, from sqrt_two_irrational_aux this H
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end
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