working

Nevanlinna/specialFunctions_Integrals.lean

@@ -8,12 +8,142 @@ 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 log volume 0 1 := by sorry
+  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
 
-  apply IntervalIntegrable.mono_fun' (g := log)
 
+  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))
@@ -24,32 +154,42 @@ example : IntervalIntegrable (log ∘ sin) volume 0 1 := by
   intro x hx
   by_contra contra
   simp at contra
-  rw [contra] at hx
-  rw [Set.left_mem_uIoc] at hx
+  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⟩
-  --
-  exact measurableSet_uIoc
-  --
 
-  have : ∀ x ∈ (Ι 0 1), ‖(log ∘ sin) x‖ ≤ log x := by sorry
+  -- 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
-  exact this
+  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
 
-  --intro x
-  rw [MeasureTheory.ae_iff]
-  simp
-
-  rw [MeasureTheory.ae_iff]
-  simp
-
-
-  sorry
+  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