81 lines
3.1 KiB
Text
81 lines
3.1 KiB
Text
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/-
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Copyright (c) 2015 Robert Y. Lewis. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Robert Y. Lewis, Jeremy Avigad
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The square root function.
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-/
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import .ivt
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open analysis real classical
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noncomputable theory
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private definition sqr_lb (x : ℝ) : ℝ := 0
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private theorem sqr_lb_is_lb (x : ℝ) (H : x ≥ 0) : (sqr_lb x) * (sqr_lb x) ≤ x :=
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by rewrite [↑sqr_lb, zero_mul]; assumption
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private definition sqr_ub (x : ℝ) : ℝ := x + 1
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private theorem sqr_ub_is_ub (x : ℝ) (H : x ≥ 0) : (sqr_ub x) * (sqr_ub x) ≥ x :=
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begin
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rewrite [↑sqr_ub, left_distrib, mul_one, right_distrib, one_mul, {x + 1}add.comm, -*add.assoc],
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apply le_add_of_nonneg_left,
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repeat apply add_nonneg,
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apply mul_nonneg,
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repeat assumption,
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apply zero_le_one
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end
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private theorem lb_le_ub (x : ℝ) (H : x ≥ 0) : sqr_lb x ≤ sqr_ub x :=
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begin
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rewrite [↑sqr_lb, ↑sqr_ub],
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apply add_nonneg,
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assumption,
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apply zero_le_one
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end
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private lemma sqr_cts : continuous (λ x : ℝ, x * x) := continuous_mul_of_continuous id_continuous id_continuous
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definition sqrt (x : ℝ) : ℝ :=
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if H : x ≥ 0 then
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some (intermediate_value_incr_weak sqr_cts (lb_le_ub x H) (sqr_lb_is_lb x H) (sqr_ub_is_ub x H))
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else 0
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private theorem sqrt_spec {x : ℝ} (H : x ≥ 0) : sqrt x * sqrt x = x ∧ sqrt x ≥ 0 :=
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begin
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rewrite [↑sqrt, dif_pos H],
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note Hs := some_spec (intermediate_value_incr_weak sqr_cts (lb_le_ub x H)
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(sqr_lb_is_lb x H) (sqr_ub_is_ub x H)),
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cases Hs with Hs1 Hs2,
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cases Hs2 with Hs2a Hs2b,
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exact and.intro Hs2b Hs1
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end
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theorem sqrt_mul_self {x : ℝ} (H : x ≥ 0) : sqrt x * sqrt x = x := and.left (sqrt_spec H)
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theorem sqrt_nonneg (x : ℝ) : sqrt x ≥ 0 :=
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if H : x ≥ 0 then and.right (sqrt_spec H) else by rewrite [↑sqrt, dif_neg H]; exact le.refl 0
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theorem sqrt_squared {x : ℝ} (H : x ≥ 0) : (sqrt x)^2 = x :=
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by krewrite [pow_two, sqrt_mul_self H]
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theorem sqrt_zero : sqrt (0 : ℝ) = 0 :=
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have sqrt 0 * sqrt 0 = 0, from sqrt_mul_self !le.refl,
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or.elim (eq_zero_or_eq_zero_of_mul_eq_zero this) (λ H, H) (λ H, H)
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theorem sqrt_squared_of_nonneg {x : ℝ} (H : x ≥ 0) : sqrt (x^2) = x :=
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have sqrt (x^2)^2 = x^2, from sqrt_squared (squared_nonneg x),
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eq_of_squared_eq_squared_of_nonneg (sqrt_nonneg (x^2)) H this
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theorem sqrt_squared' (x : ℝ) : sqrt (x^2) = abs x :=
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have x^2 = (abs x)^2, by krewrite [+pow_two, -abs_mul, abs_mul_self],
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using this, by rewrite [this, sqrt_squared_of_nonneg (abs_nonneg x)]
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theorem sqrt_mul {x y : ℝ} (Hx : x ≥ 0) (Hy : y ≥ 0) : sqrt (x * y) = sqrt x * sqrt y :=
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have (sqrt (x * y))^2 = (sqrt x * sqrt y)^2, from calc
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(sqrt (x * y))^2 = x * y : by rewrite [sqrt_squared (mul_nonneg Hx Hy)]
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... = (sqrt x)^2 * (sqrt y)^2 : by rewrite [sqrt_squared Hx, sqrt_squared Hy]
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... = (sqrt x * sqrt y)^2 : by krewrite [*pow_two]; rewrite [*mul.assoc,
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mul.left_comm (sqrt y)],
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eq_of_squared_eq_squared_of_nonneg !sqrt_nonneg (mul_nonneg !sqrt_nonneg !sqrt_nonneg) this
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