Update specialFunctions_Integrals.lean

Nevanlinna/specialFunctions_Integrals.lean

@@ -9,7 +9,6 @@ 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
@@ -18,8 +17,21 @@ lemma logsinBound : ∀ x ∈ (Set.Icc 0 1), ‖(log ∘ sin) x‖ ≤ ‖log ((
 
   -- 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
+    apply log_nonpos
-    sorry
+    apply mul_nonneg
+    apply le_of_lt
+    apply inv_pos.2
+    apply div_pos
+    exact pi_pos
+    exact zero_lt_two
+    apply (Set.mem_Icc.1 hx).1
+    simp
+    apply mul_le_one
+    rw [div_le_one pi_pos]
+    exact two_le_pi
+    exact (Set.mem_Icc.1 hx).1
+    exact (Set.mem_Icc.1 hx).2
+
   have l₁ : 0 ≤ sin x := by
     apply sin_nonneg_of_nonneg_of_le_pi (Set.mem_Icc.1 hx).1
     trans (1 : ℝ)
@@ -38,24 +50,6 @@ lemma logsinBound : ∀ x ∈ (Set.Icc 0 1), ‖(log ∘ sin) x‖ ≤ ‖log ((
     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
@@ -109,7 +103,7 @@ lemma logsinBound : ∀ x ∈ (Set.Icc 0 1), ‖(log ∘ sin) x‖ ≤ ‖log ((
     exact one_le_pi_div_two
 
 
-example : IntervalIntegrable (log ∘ sin) volume 0 1 := by
+lemma intervalIntegrable_log_sin₁ : IntervalIntegrable (log ∘ sin) volume 0 1 := by
 
   have int_log : IntervalIntegrable (fun x ↦ ‖log x‖) volume 0 1 := by
     apply IntervalIntegrable.norm
@@ -191,6 +185,47 @@ example : IntervalIntegrable (log ∘ sin) volume 0 1 := by
   exact measurable_log
   exact measurable_const_mul (π / 2)⁻¹
 
+lemma intervalIntegrable_log_sin₂ : IntervalIntegrable (log ∘ sin) volume 0 (π / 2) := by
+
+  apply IntervalIntegrable.trans (b := 1)
+  exact intervalIntegrable_log_sin₁
+
+  -- IntervalIntegrable (log ∘ sin) volume 1 (π / 2)
+  apply ContinuousOn.intervalIntegrable
+  apply ContinuousOn.comp continuousOn_log continuousOn_sin
+  intro x hx
+  rw [Set.uIcc_of_le, Set.mem_Icc] at hx
+  have : 0 < sin x := by
+    apply Real.sin_pos_of_pos_of_lt_pi
+    · calc 0
+      _ < 1 := Real.zero_lt_one
+      _ ≤ x := hx.1
+    · calc x
+      _ ≤ π / 2 := hx.2
+      _ < π := div_two_lt_of_pos pi_pos
+  by_contra h₁x
+  simp at h₁x
+  rw [h₁x] at this
+  simp at this
+  exact one_le_pi_div_two
+
+lemma intervalIntegrable_log_sin : IntervalIntegrable (log ∘ sin) volume 0 π := by
+  apply IntervalIntegrable.trans (b := π / 2)
+  exact intervalIntegrable_log_sin₂
+  let A := IntervalIntegrable.comp_sub_left intervalIntegrable_log_sin₂ π
+  simp at A
+  let B := IntervalIntegrable.symm A
+  have : π - π / 2 = π / 2 := by linarith
+  rwa [this] at B
+
+lemma intervalIntegrable_log_cos : IntervalIntegrable (log ∘ cos) volume 0 (π / 2) := by
+  let A := IntervalIntegrable.comp_sub_left intervalIntegrable_log_sin₂ (π / 2)
+  simp only [Function.comp_apply, sub_zero, sub_self] at A
+  simp_rw [sin_pi_div_two_sub] at A
+  have : (fun x => log (cos x)) = log ∘ cos := rfl
+  apply IntervalIntegrable.symm
+  rwa [← this]
+
 
 theorem logInt
   {t : ℝ}
@@ -248,8 +283,59 @@ theorem logInt
     simp [Set.mem_uIcc, ht]
 
 
-lemma int₁₁ : ∫ (x : ℝ) in (0)..π, log (4 * sin x ^ 2) = 0 := by
+lemma integral_log_sin : ∫ (x : ℝ) in (0)..(π / 2), log (sin x) =  -log 2 * π/2 := by
 
+  have t₀ {x : ℝ} : sin (2 * x) = 2 * sin x * cos x := sin_two_mul x
+
+  have t₁ {x : ℝ} : log (sin (2 * x)) = log 2 + log (sin x) + log (cos x) := by
+    rw [sin_two_mul x, log_mul, log_mul]
+    exact Ne.symm (NeZero.ne' 2)
+    sorry
+    sorry
+    sorry
+
+  have t₂ {x : ℝ} : log (sin x) = log (sin (2 * x)) - log 2 - log (cos x) := by
+    rw [t₁]
+    ring
+
+  conv =>
+    left
+    arg 1
+    intro x
+    rw [t₂]
+
+  rw [intervalIntegral.integral_sub, intervalIntegral.integral_sub]
+  rw [intervalIntegral.integral_const]
+  rw [intervalIntegral.integral_comp_mul_left (c := 2) (f := fun x ↦ log (sin x))]
+  simp
+  have : 2 * (π / 2) = π := by linarith
+  rw [this]
+
+  have : ∫ (x : ℝ) in (0)..π, log (sin x) = 2 * ∫ (x : ℝ) in (0)..(π / 2), log (sin x) := by
+    sorry
+  rw [this]
+  have : ∫ (x : ℝ) in (0)..(π / 2), log (sin x) = ∫ (x : ℝ) in (0)..(π / 2), log (cos x) := by
+    sorry
+  rw [← this]
+  simp
+  linarith
+
+  exact Ne.symm (NeZero.ne' 2)
+  -- IntervalIntegrable (fun x => log (sin (2 * x))) volume 0 (π / 2)
+  sorry
+  -- IntervalIntegrable (fun x => log 2) volume 0 (π / 2)
+  simp
+  -- IntervalIntegrable (fun x => log (sin (2 * x)) - log 2) volume 0 (π / 2)
+  apply IntervalIntegrable.sub
+  -- -- IntervalIntegrable (fun x => log (sin (2 * x))) volume 0 (π / 2)
+  sorry
+  -- -- IntervalIntegrable (fun x => log 2) volume 0 (π / 2)
+  simp
+  -- -- IntervalIntegrable (fun x => log (cos x)) volume 0 (π / 2)
+  exact intervalIntegrable_log_cos
+
+
+lemma int₁₁ : ∫ (x : ℝ) in (0)..π, log (4 * sin x ^ 2) = 0 := by
   sorry