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/- | ||
Copyright (c) 2021 Kexing Ying. All rights reserved. | ||
Released under Apache 2.0 license as described in the file LICENSE. | ||
Authors: Kexing Ying, Eric Wieser | ||
! This file was ported from Lean 3 source module data.real.sign | ||
! leanprover-community/mathlib commit 9003f28797c0664a49e4179487267c494477d853 | ||
! Please do not edit these lines, except to modify the commit id | ||
! if you have ported upstream changes. | ||
-/ | ||
import Mathlib.Data.Real.Basic | ||
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/-! | ||
# Real sign function | ||
This file introduces and contains some results about `Real.sign` which maps negative | ||
real numbers to -1, positive real numbers to 1, and 0 to 0. | ||
## Main definitions | ||
* `Real.sign r` is $\begin{cases} -1 & \text{if } r < 0, \\ | ||
~~\, 0 & \text{if } r = 0, \\ | ||
~~\, 1 & \text{if } r > 0. \end{cases}$ | ||
## Tags | ||
sign function | ||
-/ | ||
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namespace Real | ||
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/-- The sign function that maps negative real numbers to -1, positive numbers to 1, and 0 | ||
otherwise. -/ | ||
noncomputable def sign (r : ℝ) : ℝ := | ||
if r < 0 then -1 else if 0 < r then 1 else 0 | ||
#align real.sign Real.sign | ||
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theorem sign_of_neg {r : ℝ} (hr : r < 0) : sign r = -1 := by rw [sign, if_pos hr] | ||
#align real.sign_of_neg Real.sign_of_neg | ||
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theorem sign_of_pos {r : ℝ} (hr : 0 < r) : sign r = 1 := by rw [sign, if_pos hr, if_neg hr.not_lt] | ||
#align real.sign_of_pos Real.sign_of_pos | ||
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@[simp] | ||
theorem sign_zero : sign 0 = 0 := by rw [sign, if_neg (lt_irrefl _), if_neg (lt_irrefl _)] | ||
#align real.sign_zero Real.sign_zero | ||
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@[simp] | ||
theorem sign_one : sign 1 = 1 := | ||
sign_of_pos <| by norm_num | ||
#align real.sign_one Real.sign_one | ||
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theorem sign_apply_eq (r : ℝ) : sign r = -1 ∨ sign r = 0 ∨ sign r = 1 := by | ||
obtain hn | rfl | hp := lt_trichotomy r (0 : ℝ) | ||
· exact Or.inl <| sign_of_neg hn | ||
· exact Or.inr <| Or.inl <| sign_zero | ||
· exact Or.inr <| Or.inr <| sign_of_pos hp | ||
#align real.sign_apply_eq Real.sign_apply_eq | ||
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/-- This lemma is useful for working with `ℝˣ` -/ | ||
theorem sign_apply_eq_of_ne_zero (r : ℝ) (h : r ≠ 0) : sign r = -1 ∨ sign r = 1 := by | ||
obtain hn | rfl | hp := lt_trichotomy r (0 : ℝ) | ||
· exact Or.inl <| sign_of_neg hn | ||
· exact (h rfl).elim | ||
· exact Or.inr <| sign_of_pos hp | ||
#align real.sign_apply_eq_of_ne_zero Real.sign_apply_eq_of_ne_zero | ||
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@[simp] | ||
theorem sign_eq_zero_iff {r : ℝ} : sign r = 0 ↔ r = 0 := by | ||
refine' ⟨fun h => _, fun h => h.symm ▸ sign_zero⟩ | ||
obtain hn | rfl | hp := lt_trichotomy r (0 : ℝ) | ||
· rw [sign_of_neg hn, neg_eq_zero] at h | ||
exact (one_ne_zero h).elim | ||
· rfl | ||
· rw [sign_of_pos hp] at h | ||
exact (one_ne_zero h).elim | ||
#align real.sign_eq_zero_iff Real.sign_eq_zero_iff | ||
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theorem sign_int_cast (z : ℤ) : sign (z : ℝ) = ↑(Int.sign z) := by | ||
obtain hn | rfl | hp := lt_trichotomy z (0 : ℤ) | ||
· rw [sign_of_neg (Int.cast_lt_zero.mpr hn), Int.sign_eq_neg_one_of_neg hn, Int.cast_neg, | ||
Int.cast_one] | ||
· rw [Int.cast_zero, sign_zero, Int.sign_zero, Int.cast_zero] | ||
· rw [sign_of_pos (Int.cast_pos.mpr hp), Int.sign_eq_one_of_pos hp, Int.cast_one] | ||
#align real.sign_int_cast Real.sign_int_cast | ||
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theorem sign_neg {r : ℝ} : sign (-r) = -sign r := by | ||
obtain hn | rfl | hp := lt_trichotomy r (0 : ℝ) | ||
· rw [sign_of_neg hn, sign_of_pos (neg_pos.mpr hn), neg_neg] | ||
· rw [sign_zero, neg_zero, sign_zero] | ||
· rw [sign_of_pos hp, sign_of_neg (neg_lt_zero.mpr hp)] | ||
#align real.sign_neg Real.sign_neg | ||
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theorem sign_mul_nonneg (r : ℝ) : 0 ≤ sign r * r := by | ||
obtain hn | rfl | hp := lt_trichotomy r (0 : ℝ) | ||
· rw [sign_of_neg hn] | ||
exact mul_nonneg_of_nonpos_of_nonpos (by norm_num) hn.le | ||
· rw [mul_zero] | ||
· rw [sign_of_pos hp, one_mul] | ||
exact hp.le | ||
#align real.sign_mul_nonneg Real.sign_mul_nonneg | ||
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theorem sign_mul_pos_of_ne_zero (r : ℝ) (hr : r ≠ 0) : 0 < sign r * r := by | ||
refine' lt_of_le_of_ne (sign_mul_nonneg r) fun h => hr _ | ||
have hs0 := (zero_eq_mul.mp h).resolve_right hr | ||
exact sign_eq_zero_iff.mp hs0 | ||
#align real.sign_mul_pos_of_ne_zero Real.sign_mul_pos_of_ne_zero | ||
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@[simp] | ||
theorem inv_sign (r : ℝ) : (sign r)⁻¹ = sign r := by | ||
obtain hn | hz | hp := sign_apply_eq r | ||
· rw [hn] | ||
norm_num | ||
· rw [hz] | ||
exact inv_zero | ||
· rw [hp] | ||
exact inv_one | ||
#align real.inv_sign Real.inv_sign | ||
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@[simp] | ||
theorem sign_inv (r : ℝ) : sign r⁻¹ = sign r := by | ||
obtain hn | rfl | hp := lt_trichotomy r (0 : ℝ) | ||
· rw [sign_of_neg hn, sign_of_neg (inv_lt_zero.mpr hn)] | ||
· rw [sign_zero, inv_zero, sign_zero] | ||
· rw [sign_of_pos hp, sign_of_pos (inv_pos.mpr hp)] | ||
#align real.sign_inv Real.sign_inv | ||
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end Real |