303 lines
9.5 KiB
Plaintext
303 lines
9.5 KiB
Plaintext
import Mathlib.Analysis.SpecialFunctions.Integrals
|
||
import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
|
||
import Mathlib.MeasureTheory.Integral.CircleIntegral
|
||
import Mathlib.MeasureTheory.Measure.Restrict
|
||
|
||
open scoped Interval Topology
|
||
open Real Filter MeasureTheory intervalIntegral
|
||
|
||
-- The following theorem was suggested by Gareth Ma on Zulip
|
||
|
||
|
||
|
||
lemma logsinBound : ∀ x ∈ (Set.Icc 0 1), ‖(log ∘ sin) x‖ ≤ ‖log ((π / 2)⁻¹ * x)‖ := by
|
||
|
||
intro x hx
|
||
by_cases h'x : x = 0
|
||
· rw [h'x]; simp
|
||
|
||
-- Now handle the case where x ≠ 0
|
||
have l₀ : log ((π / 2)⁻¹ * x) ≤ 0 := by
|
||
-- log_nonpos (Set.mem_Icc.1 hx).1 (Set.mem_Icc.1 hx).2
|
||
sorry
|
||
have l₁ : 0 ≤ sin x := by
|
||
apply sin_nonneg_of_nonneg_of_le_pi (Set.mem_Icc.1 hx).1
|
||
trans (1 : ℝ)
|
||
exact (Set.mem_Icc.1 hx).2
|
||
trans π / 2
|
||
exact one_le_pi_div_two
|
||
norm_num [pi_nonneg]
|
||
have l₂ : log (sin x) ≤ 0 := log_nonpos l₁ (sin_le_one x)
|
||
|
||
simp only [norm_eq_abs, Function.comp_apply]
|
||
rw [abs_eq_neg_self.2 l₀]
|
||
rw [abs_eq_neg_self.2 l₂]
|
||
simp only [neg_le_neg_iff, ge_iff_le]
|
||
|
||
have l₃ : x ∈ (Set.Ioi 0) := by
|
||
simp
|
||
exact lt_of_le_of_ne (Set.mem_Icc.1 hx).1 ( fun a => h'x (id (Eq.symm a)) )
|
||
|
||
have l₄ : sin x ∈ (Set.Ioi 0) := by
|
||
have t₁ : 0 ∈ Set.Icc (-(π / 2)) (π / 2) := by
|
||
simp
|
||
apply div_nonneg pi_nonneg zero_le_two
|
||
have t₂ : x ∈ Set.Icc (-(π / 2)) (π / 2) := by
|
||
simp
|
||
constructor
|
||
· trans 0
|
||
simp
|
||
apply div_nonneg pi_nonneg zero_le_two
|
||
exact (Set.mem_Icc.1 hx).1
|
||
· trans (1 : ℝ)
|
||
exact (Set.mem_Icc.1 hx).2
|
||
exact one_le_pi_div_two
|
||
let A := Real.strictMonoOn_sin t₁ t₂ l₃
|
||
simp at A
|
||
simpa
|
||
|
||
have l₅ : 0 < (π / 2)⁻¹ * x := by
|
||
apply mul_pos
|
||
apply inv_pos.2
|
||
apply div_pos pi_pos zero_lt_two
|
||
exact l₃
|
||
|
||
have : ∀ x ∈ (Set.Icc 0 (π / 2)), (π / 2)⁻¹ * x ≤ sin x := by
|
||
intro x hx
|
||
|
||
have i₀ : 0 ∈ Set.Icc 0 π :=
|
||
Set.left_mem_Icc.mpr pi_nonneg
|
||
have i₁ : π / 2 ∈ Set.Icc 0 π :=
|
||
Set.mem_Icc.mpr ⟨div_nonneg pi_nonneg zero_le_two, half_le_self pi_nonneg⟩
|
||
|
||
have i₂ : 0 ≤ 1 - (π / 2)⁻¹ * x := by
|
||
rw [sub_nonneg]
|
||
calc (π / 2)⁻¹ * x
|
||
_ ≤ (π / 2)⁻¹ * (π / 2) := by
|
||
apply mul_le_mul_of_nonneg_left
|
||
exact (Set.mem_Icc.1 hx).2
|
||
apply inv_nonneg.mpr (div_nonneg pi_nonneg zero_le_two)
|
||
_ = 1 := by
|
||
apply inv_mul_cancel
|
||
apply div_ne_zero_iff.mpr
|
||
constructor
|
||
· exact pi_ne_zero
|
||
· exact Ne.symm (NeZero.ne' 2)
|
||
|
||
have i₃ : 0 ≤ (π / 2)⁻¹ * x := by
|
||
apply mul_nonneg
|
||
apply inv_nonneg.2
|
||
apply div_nonneg
|
||
exact pi_nonneg
|
||
exact zero_le_two
|
||
exact (Set.mem_Icc.1 hx).1
|
||
|
||
have i₄ : 1 - (π / 2)⁻¹ * x + (π / 2)⁻¹ * x = 1 := by ring
|
||
|
||
let B := strictConcaveOn_sin_Icc.concaveOn.2 i₀ i₁ i₂ i₃ i₄
|
||
simp [Real.sin_pi_div_two] at B
|
||
rw [(by ring_nf; rw [mul_inv_cancel pi_ne_zero, one_mul] : 2 / π * x * (π / 2) = x)] at B
|
||
simpa
|
||
|
||
apply log_le_log l₅
|
||
apply this
|
||
apply Set.mem_Icc.mpr
|
||
constructor
|
||
· exact le_of_lt l₃
|
||
· trans 1
|
||
exact (Set.mem_Icc.1 hx).2
|
||
exact one_le_pi_div_two
|
||
|
||
|
||
example : IntervalIntegrable (log ∘ sin) volume 0 1 := by
|
||
|
||
have int_log : IntervalIntegrable (fun x ↦ ‖log x‖) volume 0 1 := by
|
||
apply IntervalIntegrable.norm
|
||
rw [← neg_neg log]
|
||
apply IntervalIntegrable.neg
|
||
apply intervalIntegrable_deriv_of_nonneg (g := fun x ↦ -(x * log x - x))
|
||
· exact (continuous_mul_log.continuousOn.sub continuous_id.continuousOn).neg
|
||
· intro x hx
|
||
norm_num at hx
|
||
convert ((hasDerivAt_mul_log hx.left.ne.symm).sub (hasDerivAt_id x)).neg using 1
|
||
norm_num
|
||
· intro x hx
|
||
norm_num at hx
|
||
rw [Pi.neg_apply, Left.nonneg_neg_iff]
|
||
exact (log_nonpos_iff hx.left).mpr hx.right.le
|
||
|
||
|
||
have int_log : IntervalIntegrable (fun x ↦ ‖log ((π / 2)⁻¹ * x)‖) volume 0 1 := by
|
||
|
||
have A := IntervalIntegrable.comp_mul_right int_log (π / 2)⁻¹
|
||
simp only [norm_eq_abs] at A
|
||
conv =>
|
||
arg 1
|
||
intro x
|
||
rw [mul_comm]
|
||
simp only [norm_eq_abs]
|
||
apply IntervalIntegrable.mono A
|
||
simp
|
||
trans Set.Icc 0 (π / 2)
|
||
exact Set.Icc_subset_Icc (Preorder.le_refl 0) one_le_pi_div_two
|
||
exact Set.Icc_subset_uIcc
|
||
exact Preorder.le_refl volume
|
||
|
||
apply IntervalIntegrable.mono_fun' (g := fun x ↦ ‖log ((π / 2)⁻¹ * x)‖)
|
||
exact int_log
|
||
|
||
-- AEStronglyMeasurable (log ∘ sin) (volume.restrict (Ι 0 1))
|
||
apply ContinuousOn.aestronglyMeasurable
|
||
apply ContinuousOn.comp (t := Ι 0 1)
|
||
apply ContinuousOn.mono (s := {0}ᶜ)
|
||
exact continuousOn_log
|
||
intro x hx
|
||
by_contra contra
|
||
simp at contra
|
||
rw [contra, Set.left_mem_uIoc] at hx
|
||
linarith
|
||
exact continuousOn_sin
|
||
|
||
-- Set.MapsTo sin (Ι 0 1) (Ι 0 1)
|
||
rw [Set.uIoc_of_le (zero_le_one' ℝ)]
|
||
exact fun x hx ↦ ⟨sin_pos_of_pos_of_le_one hx.1 hx.2, sin_le_one x⟩
|
||
|
||
-- MeasurableSet (Ι 0 1)
|
||
exact measurableSet_uIoc
|
||
|
||
-- (fun x => ‖(log ∘ sin) x‖) ≤ᶠ[ae (volume.restrict (Ι 0 1))] ‖log‖
|
||
dsimp [EventuallyLE]
|
||
rw [MeasureTheory.ae_restrict_iff]
|
||
apply MeasureTheory.ae_of_all
|
||
intro x hx
|
||
have : x ∈ Set.Icc 0 1 := by
|
||
simp
|
||
simp at hx
|
||
constructor
|
||
· exact le_of_lt hx.1
|
||
· exact hx.2
|
||
let A := logsinBound x this
|
||
simp only [Function.comp_apply, norm_eq_abs] at A
|
||
exact A
|
||
|
||
apply measurableSet_le
|
||
apply Measurable.comp'
|
||
exact continuous_abs.measurable
|
||
exact Measurable.comp' measurable_log continuous_sin.measurable
|
||
-- Measurable fun a => |log ((π / 2)⁻¹ * a)|
|
||
apply Measurable.comp'
|
||
exact continuous_abs.measurable
|
||
apply Measurable.comp'
|
||
exact measurable_log
|
||
exact measurable_const_mul (π / 2)⁻¹
|
||
|
||
|
||
theorem logInt
|
||
{t : ℝ}
|
||
(ht : 0 < t) :
|
||
∫ x in (0 : ℝ)..t, log x = t * log t - t := by
|
||
rw [← integral_add_adjacent_intervals (b := 1)]
|
||
trans (-1) + (t * log t - t + 1)
|
||
· congr
|
||
· -- ∫ x in 0..1, log x = -1, same proof as before
|
||
rw [integral_eq_sub_of_hasDerivAt_of_tendsto (f := fun x ↦ x * log x - x) (fa := 0) (fb := -1)]
|
||
· simp
|
||
· simp
|
||
· intro x hx
|
||
norm_num at hx
|
||
convert (hasDerivAt_mul_log hx.left.ne.symm).sub (hasDerivAt_id x) using 1
|
||
norm_num
|
||
· rw [← neg_neg log]
|
||
apply IntervalIntegrable.neg
|
||
apply intervalIntegrable_deriv_of_nonneg (g := fun x ↦ -(x * log x - x))
|
||
· exact (continuous_mul_log.continuousOn.sub continuous_id.continuousOn).neg
|
||
· intro x hx
|
||
norm_num at hx
|
||
convert ((hasDerivAt_mul_log hx.left.ne.symm).sub (hasDerivAt_id x)).neg using 1
|
||
norm_num
|
||
· intro x hx
|
||
norm_num at hx
|
||
rw [Pi.neg_apply, Left.nonneg_neg_iff]
|
||
exact (log_nonpos_iff hx.left).mpr hx.right.le
|
||
· have := tendsto_log_mul_rpow_nhds_zero zero_lt_one
|
||
simp_rw [rpow_one, mul_comm] at this
|
||
-- tendsto_nhdsWithin_of_tendsto_nhds should be under Tendsto namespace
|
||
convert this.sub (tendsto_nhdsWithin_of_tendsto_nhds tendsto_id)
|
||
norm_num
|
||
· rw [(by simp : -1 = 1 * log 1 - 1)]
|
||
apply tendsto_nhdsWithin_of_tendsto_nhds
|
||
exact (continuousAt_id.mul (continuousAt_log one_ne_zero)).sub continuousAt_id
|
||
· -- ∫ x in 1..t, log x = t * log t + 1, just use integral_log_of_pos
|
||
rw [integral_log_of_pos zero_lt_one ht]
|
||
norm_num
|
||
· abel
|
||
· -- log is integrable on [[0, 1]]
|
||
rw [← neg_neg log]
|
||
apply IntervalIntegrable.neg
|
||
apply intervalIntegrable_deriv_of_nonneg (g := fun x ↦ -(x * log x - x))
|
||
· exact (continuous_mul_log.continuousOn.sub continuous_id.continuousOn).neg
|
||
· intro x hx
|
||
norm_num at hx
|
||
convert ((hasDerivAt_mul_log hx.left.ne.symm).sub (hasDerivAt_id x)).neg using 1
|
||
norm_num
|
||
· intro x hx
|
||
norm_num at hx
|
||
rw [Pi.neg_apply, Left.nonneg_neg_iff]
|
||
exact (log_nonpos_iff hx.left).mpr hx.right.le
|
||
· -- log is integrable on [[0, t]]
|
||
simp [Set.mem_uIcc, ht]
|
||
|
||
|
||
lemma int₁₁ : ∫ (x : ℝ) in (0)..π, log (4 * sin x ^ 2) = 0 := by
|
||
|
||
sorry
|
||
|
||
|
||
lemma int₁ :
|
||
∫ x in (0)..(2 * π), log ‖circleMap 0 1 x - 1‖ = 0 := by
|
||
|
||
have {x : ℝ} : log ‖circleMap 0 1 x - 1‖ = log (4 * sin (x / 2) ^ 2) / 2 := by
|
||
dsimp [Complex.abs]
|
||
rw [log_sqrt (Complex.normSq_nonneg (circleMap 0 1 x - 1))]
|
||
congr
|
||
calc Complex.normSq (circleMap 0 1 x - 1)
|
||
_ = (cos x - 1) * (cos x - 1) + sin x * sin x := by
|
||
dsimp [circleMap]
|
||
rw [Complex.normSq_apply]
|
||
simp
|
||
_ = sin x ^ 2 + cos x ^ 2 + 1 - 2 * cos x := by
|
||
ring
|
||
_ = 2 - 2 * cos x := by
|
||
rw [sin_sq_add_cos_sq]
|
||
norm_num
|
||
_ = 2 - 2 * cos (2 * (x / 2)) := by
|
||
rw [← mul_div_assoc]
|
||
congr; norm_num
|
||
_ = 4 - 4 * Real.cos (x / 2) ^ 2 := by
|
||
rw [cos_two_mul]
|
||
ring
|
||
_ = 4 * sin (x / 2) ^ 2 := by
|
||
nth_rw 1 [← mul_one 4, ← sin_sq_add_cos_sq (x / 2)]
|
||
ring
|
||
simp_rw [this]
|
||
simp
|
||
|
||
have : ∫ (x : ℝ) in (0)..2 * π, log (4 * sin (x / 2) ^ 2) = 2 * ∫ (x : ℝ) in (0)..π, log (4 * sin x ^ 2) := by
|
||
have : 1 = 2 * (2 : ℝ)⁻¹ := by exact Eq.symm (mul_inv_cancel_of_invertible 2)
|
||
nth_rw 1 [← one_mul (∫ (x : ℝ) in (0)..2 * π, log (4 * sin (x / 2) ^ 2))]
|
||
rw [← mul_inv_cancel_of_invertible 2, mul_assoc]
|
||
let f := fun y ↦ log (4 * sin y ^ 2)
|
||
have {x : ℝ} : log (4 * sin (x / 2) ^ 2) = f (x / 2) := by simp
|
||
conv =>
|
||
left
|
||
right
|
||
right
|
||
arg 1
|
||
intro x
|
||
rw [this]
|
||
rw [intervalIntegral.inv_mul_integral_comp_div 2]
|
||
simp
|
||
rw [this]
|
||
simp
|
||
exact int₁₁
|