feat: port Analysis.SpecialFunctions.Trigonometric.Bounds (#4768)

Mathlib.lean

@@ -657,6 +657,7 @@ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
 import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan
 import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
 import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
+import Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds
 import Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev
 import Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex
 import Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv

Mathlib/Analysis/SpecialFunctions/Trigonometric/Bounds.lean

@@ -0,0 +1,160 @@
+/-
+Copyright (c) 2022 David Loeffler. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: David Loeffler
+
+! This file was ported from Lean 3 source module analysis.special_functions.trigonometric.bounds
+! leanprover-community/mathlib commit 2c1d8ca2812b64f88992a5294ea3dba144755cd1
+! Please do not edit these lines, except to modify the commit id
+! if you have ported upstream changes.
+-/
+import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
+
+/-!
+# Polynomial bounds for trigonometric functions
+
+## Main statements
+
+This file contains upper and lower bounds for real trigonometric functions in terms
+of polynomials. See `Trigonometric.Basic` for more elementary inequalities, establishing
+the ranges of these functions, and their monotonicity in suitable intervals.
+
+Here we prove the following:
+
+* `sin_lt`: for `x > 0` we have `sin x < x`.
+* `sin_gt_sub_cube`: For `0 < x ≤ 1` we have `x - x ^ 3 / 4 < sin x`.
+* `lt_tan`: for `0 < x < π/2` we have `x < tan x`.
+* `cos_le_one_div_sqrt_sq_add_one` and `cos_lt_one_div_sqrt_sq_add_one`: for
+  `-3 * π / 2 ≤ x ≤ 3 * π / 2`, we have `cos x ≤ 1 / sqrt (x ^ 2 + 1)`, with strict inequality if
+  `x ≠ 0`. (This bound is not quite optimal, but not far off)
+
+## Tags
+
+sin, cos, tan, angle
+-/
+
+
+noncomputable section
+
+local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y) -- Porting note: See issue #2220
+
+open Set
+
+namespace Real
+
+open scoped Real
+
+/-- For 0 < x, we have sin x < x. -/
+theorem sin_lt {x : ℝ} (h : 0 < x) : sin x < x := by
+  cases' lt_or_le 1 x with h' h'
+  · exact (sin_le_one x).trans_lt h'
+  have hx : |x| = x := abs_of_nonneg h.le
+  have := le_of_abs_le (sin_bound <| show |x| ≤ 1 by rwa [hx])
+  rw [sub_le_iff_le_add', hx] at this
+  apply this.trans_lt
+  rw [sub_add, sub_lt_self_iff, sub_pos, div_eq_mul_inv (x ^ 3)]
+  refine' mul_lt_mul' _ (by norm_num) (by norm_num) (pow_pos h 3)
+  apply pow_le_pow_of_le_one h.le h'
+  norm_num
+#align real.sin_lt Real.sin_lt
+
+/-- For 0 < x ≤ 1 we have x - x ^ 3 / 4 < sin x.
+
+This is also true for x > 1, but it's nontrivial for x just above 1. This inequality is not
+tight; the tighter inequality is sin x > x - x ^ 3 / 6 for all x > 0, but this inequality has
+a simpler proof. -/
+theorem sin_gt_sub_cube {x : ℝ} (h : 0 < x) (h' : x ≤ 1) : x - x ^ 3 / 4 < sin x := by
+  have hx : |x| = x := abs_of_nonneg h.le
+  have := neg_le_of_abs_le (sin_bound <| show |x| ≤ 1 by rwa [hx])
+  rw [le_sub_iff_add_le, hx] at this
+  refine' lt_of_lt_of_le _ this
+  have : x ^ 3 / ↑4 - x ^ 3 / ↑6 = x ^ 3 * 12⁻¹ := by ring
+  rw [add_comm, sub_add, sub_neg_eq_add, sub_lt_sub_iff_left, ← lt_sub_iff_add_lt', this]
+  refine' mul_lt_mul' _ (by norm_num) (by norm_num) (pow_pos h 3)
+  apply pow_le_pow_of_le_one h.le h'
+  norm_num
+#align real.sin_gt_sub_cube Real.sin_gt_sub_cube
+
+/-- The derivative of `tan x - x` is `1/(cos x)^2 - 1` away from the zeroes of cos. -/
+theorem deriv_tan_sub_id (x : ℝ) (h : cos x ≠ 0) :
+    deriv (fun y : ℝ => tan y - y) x = 1 / cos x ^ 2 - 1 :=
+  HasDerivAt.deriv <| by simpa using (hasDerivAt_tan h).add (hasDerivAt_id x).neg
+#align real.deriv_tan_sub_id Real.deriv_tan_sub_id
+
+/-- For all `0 < x < π/2` we have `x < tan x`.
+
+This is proved by checking that the function `tan x - x` vanishes
+at zero and has non-negative derivative. -/
+theorem lt_tan {x : ℝ} (h1 : 0 < x) (h2 : x < π / 2) : x < tan x := by
+  let U := Ico 0 (π / 2)
+  have intU : interior U = Ioo 0 (π / 2) := interior_Ico
+  have half_pi_pos : 0 < π / 2 := div_pos pi_pos two_pos
+  have cos_pos : ∀ {y : ℝ}, y ∈ U → 0 < cos y := by
+    intro y hy
+    exact cos_pos_of_mem_Ioo (Ico_subset_Ioo_left (neg_lt_zero.mpr half_pi_pos) hy)
+  have sin_pos : ∀ {y : ℝ}, y ∈ interior U → 0 < sin y := by
+    intro y hy
+    rw [intU] at hy
+    exact sin_pos_of_mem_Ioo (Ioo_subset_Ioo_right (div_le_self pi_pos.le one_le_two) hy)
+  have tan_cts_U : ContinuousOn tan U := by
+    apply ContinuousOn.mono continuousOn_tan
+    intro z hz
+    simp only [mem_setOf_eq]
+    exact (cos_pos hz).ne'
+  have tan_minus_id_cts : ContinuousOn (fun y : ℝ => tan y - y) U := tan_cts_U.sub continuousOn_id
+  have deriv_pos : ∀ y : ℝ, y ∈ interior U → 0 < deriv (fun y' : ℝ => tan y' - y') y := by
+    intro y hy
+    have := cos_pos (interior_subset hy)
+    simp only [deriv_tan_sub_id y this.ne', one_div, gt_iff_lt, sub_pos]
+    norm_cast
+    have bd2 : cos y ^ 2 < 1 := by
+      apply lt_of_le_of_ne y.cos_sq_le_one
+      rw [cos_sq']
+      simpa only [Ne.def, sub_eq_self, pow_eq_zero_iff, Nat.succ_pos'] using (sin_pos hy).ne'
+    rwa [one_div, lt_inv, inv_one]
+    · exact zero_lt_one
+    simpa only [sq, mul_self_pos] using this.ne'
+  have mono := Convex.strictMonoOn_of_deriv_pos (convex_Ico 0 (π / 2)) tan_minus_id_cts deriv_pos
+  have zero_in_U : (0 : ℝ) ∈ U := by rwa [left_mem_Ico]
+  have x_in_U : x ∈ U := ⟨h1.le, h2⟩
+  simpa only [tan_zero, sub_zero, sub_pos] using mono zero_in_U x_in_U h1
+#align real.lt_tan Real.lt_tan
+
+theorem le_tan {x : ℝ} (h1 : 0 ≤ x) (h2 : x < π / 2) : x ≤ tan x := by
+  rcases eq_or_lt_of_le h1 with (rfl | h1')
+  · rw [tan_zero]
+  · exact le_of_lt (lt_tan h1' h2)
+#align real.le_tan Real.le_tan
+
+theorem cos_lt_one_div_sqrt_sq_add_one {x : ℝ} (hx1 : -(3 * π / 2) ≤ x) (hx2 : x ≤ 3 * π / 2)
+    (hx3 : x ≠ 0) : cos x < ↑1 / sqrt (x ^ 2 + 1) := by
+  suffices ∀ {y : ℝ} (_ : 0 < y) (_ : y ≤ 3 * π / 2), cos y < ↑1 / sqrt (y ^ 2 + 1) by
+    rcases lt_or_lt_iff_ne.mpr hx3.symm with ⟨h⟩
+    · exact this h hx2
+    · convert this (by linarith : 0 < -x) (by linarith) using 1
+      · rw [cos_neg]
+      · rw [neg_sq]
+  intro y hy1 hy2
+  have hy3 : ↑0 < y ^ 2 + 1 := by linarith [sq_nonneg y]
+  rcases lt_or_le y (π / 2) with (hy2' | hy1')
+  · -- Main case : `0 < y < π / 2`
+    have hy4 : 0 < cos y := cos_pos_of_mem_Ioo ⟨by linarith, hy2'⟩
+    rw [← abs_of_nonneg (cos_nonneg_of_mem_Icc ⟨by linarith, hy2'.le⟩), ←
+      abs_of_nonneg (one_div_nonneg.mpr (sqrt_nonneg _)), ← sq_lt_sq, div_pow, one_pow,
+      sq_sqrt hy3.le, lt_one_div (pow_pos hy4 _) hy3, ← inv_one_add_tan_sq hy4.ne', one_div,
+      inv_inv, add_comm, add_lt_add_iff_left, sq_lt_sq, abs_of_pos hy1,
+      abs_of_nonneg (tan_nonneg_of_nonneg_of_le_pi_div_two hy1.le hy2'.le)]
+    exact Real.lt_tan hy1 hy2'
+  · -- Easy case : `π / 2 ≤ y ≤ 3 * π / 2`
+    refine' lt_of_le_of_lt _ (one_div_pos.mpr <| sqrt_pos_of_pos hy3)
+    exact cos_nonpos_of_pi_div_two_le_of_le hy1' (by linarith [pi_pos])
+#align real.cos_lt_one_div_sqrt_sq_add_one Real.cos_lt_one_div_sqrt_sq_add_one
+
+theorem cos_le_one_div_sqrt_sq_add_one {x : ℝ} (hx1 : -(3 * π / 2) ≤ x) (hx2 : x ≤ 3 * π / 2) :
+    cos x ≤ ↑1 / sqrt (x ^ 2 + 1) := by
+  rcases eq_or_ne x 0 with (rfl | hx3)
+  · simp
+  · exact (cos_lt_one_div_sqrt_sq_add_one hx1 hx2 hx3).le
+#align real.cos_le_one_div_sqrt_sq_add_one Real.cos_le_one_div_sqrt_sq_add_one
+
+end Real