2016-02-20 18:17:10 +00:00
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/-
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Copyright (c) 2016 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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Instantiate the complex numbers as a normed space, by temporarily making it an inner product space
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over the reals.
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-/
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import theories.analysis.inner_product data.complex
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open nat real complex analysis classical
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noncomputable theory
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namespace complex
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namespace real_inner_product_space
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definition smul (a : ℝ) (z : ℂ) : ℂ := complex.mk (a * re z) (a * im z)
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definition ip (z w : ℂ) : ℝ := re z * re w + im z * im w
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proposition smul_left_distrib (a : ℝ) (z w : ℂ) : smul a (z + w) = smul a z + smul a w :=
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by rewrite [↑smul, *re_add, *im_add, *left_distrib]
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proposition smul_right_distrib (a b : ℝ) (z : ℂ) : smul (a + b) z = smul a z + smul b z :=
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by rewrite [↑smul, *right_distrib]
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proposition mul_smul (a b : ℝ) (z : ℂ) : smul (a * b) z = smul a (smul b z) :=
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by rewrite [↑smul, *mul.assoc]
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proposition one_smul (z : ℂ) : smul 1 z = z := by rewrite [↑smul, *one_mul, complex.eta]
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proposition inner_add_left (x y z : ℂ) : ip (x + y) z = ip x z + ip y z :=
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by rewrite [↑ip, re_add, im_add, *right_distrib, *add.assoc, add.left_comm (re y * re z)]
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proposition inner_smul_left (a : ℝ) (x y : ℂ) : ip (smul a x) y = a * ip x y :=
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by rewrite [↑ip, ↑smul, left_distrib, *mul.assoc]
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proposition inner_comm (x y : ℂ) : ip x y = ip y x :=
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by rewrite [↑ip, mul.comm, mul.comm (im x)]
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proposition inner_self_nonneg (x : ℂ) : ip x x ≥ 0 :=
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add_nonneg (mul_self_nonneg (re x)) (mul_self_nonneg (im x))
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proposition eq_zero_of_inner_self_eq_zero {x : ℂ} (H : ip x x = 0) : x = 0 :=
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have re x = 0, from eq_zero_of_mul_self_add_mul_self_eq_zero H,
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have im x = 0, from eq_zero_of_mul_self_add_mul_self_eq_zero
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(by rewrite [↑ip at H, add.comm at H]; exact H),
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2016-02-29 21:15:48 +00:00
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by rewrite [-complex.eta, `re x = 0`, `im x = 0`]
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2016-02-20 18:17:10 +00:00
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end real_inner_product_space
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protected definition real_inner_product_space [reducible] : inner_product_space ℂ :=
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⦃ inner_product_space, complex.discrete_field,
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smul := real_inner_product_space.smul,
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inner := real_inner_product_space.ip,
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smul_left_distrib := real_inner_product_space.smul_left_distrib,
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smul_right_distrib := real_inner_product_space.smul_right_distrib,
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mul_smul := real_inner_product_space.mul_smul,
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one_smul := real_inner_product_space.one_smul,
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inner_add_left := real_inner_product_space.inner_add_left,
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inner_smul_left := real_inner_product_space.inner_smul_left,
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inner_comm := real_inner_product_space.inner_comm,
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inner_self_nonneg := real_inner_product_space.inner_self_nonneg,
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eq_zero_of_inner_self_eq_zero := @real_inner_product_space.eq_zero_of_inner_self_eq_zero
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⦄
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local attribute complex.real_inner_product_space [trans_instance]
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2016-02-25 22:30:00 +00:00
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protected definition normed_vector_space [trans_instance] : normed_vector_space ℂ :=
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2016-02-20 18:17:10 +00:00
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_
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theorem norm_squared_eq_cmod (z : ℂ) : ∥ z ∥^2 = cmod z := by rewrite norm_squared
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end complex
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