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eb58a8df04
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3063415cf9
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@ -1,62 +0,0 @@
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import Mathlib.Analysis.SpecialFunctions.Integrals
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import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
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import Mathlib.MeasureTheory.Integral.CircleIntegral
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import Mathlib.MeasureTheory.Measure.Restrict
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open scoped Interval Topology
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open Real Filter MeasureTheory intervalIntegral
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lemma int₁₁ : ∫ (x : ℝ) in (0)..π, log (4 * sin x ^ 2) = 0 := by
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sorry
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lemma int₁ :
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∫ x in (0)..(2 * π), log ‖circleMap 0 1 x - 1‖ = 0 := by
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have {x : ℝ} : log ‖circleMap 0 1 x - 1‖ = log (4 * sin (x / 2) ^ 2) / 2 := by
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dsimp [Complex.abs]
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rw [log_sqrt (Complex.normSq_nonneg (circleMap 0 1 x - 1))]
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congr
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calc Complex.normSq (circleMap 0 1 x - 1)
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_ = (cos x - 1) * (cos x - 1) + sin x * sin x := by
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dsimp [circleMap]
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rw [Complex.normSq_apply]
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simp
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_ = sin x ^ 2 + cos x ^ 2 + 1 - 2 * cos x := by
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ring
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_ = 2 - 2 * cos x := by
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rw [sin_sq_add_cos_sq]
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norm_num
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_ = 2 - 2 * cos (2 * (x / 2)) := by
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rw [← mul_div_assoc]
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congr; norm_num
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_ = 4 - 4 * Real.cos (x / 2) ^ 2 := by
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rw [cos_two_mul]
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ring
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_ = 4 * sin (x / 2) ^ 2 := by
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nth_rw 1 [← mul_one 4, ← sin_sq_add_cos_sq (x / 2)]
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ring
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simp_rw [this]
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simp
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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
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have : 1 = 2 * (2 : ℝ)⁻¹ := by exact Eq.symm (mul_inv_cancel_of_invertible 2)
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nth_rw 1 [← one_mul (∫ (x : ℝ) in (0)..2 * π, log (4 * sin (x / 2) ^ 2))]
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rw [← mul_inv_cancel_of_invertible 2, mul_assoc]
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let f := fun y ↦ log (4 * sin y ^ 2)
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have {x : ℝ} : log (4 * sin (x / 2) ^ 2) = f (x / 2) := by simp
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conv =>
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left
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right
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right
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arg 1
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intro x
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rw [this]
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rw [intervalIntegral.inv_mul_integral_comp_div 2]
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simp
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rw [this]
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simp
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exact int₁₁
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@ -1,65 +0,0 @@
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import Mathlib.Analysis.SpecialFunctions.Integrals
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import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
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import Mathlib.MeasureTheory.Integral.CircleIntegral
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import Mathlib.MeasureTheory.Measure.Restrict
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open scoped Interval Topology
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open Real Filter MeasureTheory intervalIntegral
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-- The following theorem was suggested by Gareth Ma on Zulip
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theorem logInt
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{t : ℝ}
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(ht : 0 < t) :
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∫ x in (0 : ℝ)..t, log x = t * log t - t := by
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rw [← integral_add_adjacent_intervals (b := 1)]
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trans (-1) + (t * log t - t + 1)
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· congr
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· -- ∫ x in 0..1, log x = -1, same proof as before
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rw [integral_eq_sub_of_hasDerivAt_of_tendsto (f := fun x ↦ x * log x - x) (fa := 0) (fb := -1)]
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· simp
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· simp
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· intro x hx
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norm_num at hx
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convert (hasDerivAt_mul_log hx.left.ne.symm).sub (hasDerivAt_id x) using 1
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norm_num
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· rw [← neg_neg log]
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apply IntervalIntegrable.neg
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apply intervalIntegrable_deriv_of_nonneg (g := fun x ↦ -(x * log x - x))
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· exact (continuous_mul_log.continuousOn.sub continuous_id.continuousOn).neg
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· intro x hx
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norm_num at hx
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convert ((hasDerivAt_mul_log hx.left.ne.symm).sub (hasDerivAt_id x)).neg using 1
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norm_num
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· intro x hx
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norm_num at hx
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rw [Pi.neg_apply, Left.nonneg_neg_iff]
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exact (log_nonpos_iff hx.left).mpr hx.right.le
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· have := tendsto_log_mul_rpow_nhds_zero zero_lt_one
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simp_rw [rpow_one, mul_comm] at this
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-- tendsto_nhdsWithin_of_tendsto_nhds should be under Tendsto namespace
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convert this.sub (tendsto_nhdsWithin_of_tendsto_nhds tendsto_id)
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norm_num
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· rw [(by simp : -1 = 1 * log 1 - 1)]
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apply tendsto_nhdsWithin_of_tendsto_nhds
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exact (continuousAt_id.mul (continuousAt_log one_ne_zero)).sub continuousAt_id
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· -- ∫ x in 1..t, log x = t * log t + 1, just use integral_log_of_pos
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rw [integral_log_of_pos zero_lt_one ht]
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norm_num
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· abel
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· -- log is integrable on [[0, 1]]
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rw [← neg_neg log]
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apply IntervalIntegrable.neg
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apply intervalIntegrable_deriv_of_nonneg (g := fun x ↦ -(x * log x - x))
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· exact (continuous_mul_log.continuousOn.sub continuous_id.continuousOn).neg
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· intro x hx
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norm_num at hx
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convert ((hasDerivAt_mul_log hx.left.ne.symm).sub (hasDerivAt_id x)).neg using 1
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norm_num
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· intro x hx
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norm_num at hx
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rw [Pi.neg_apply, Left.nonneg_neg_iff]
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exact (log_nonpos_iff hx.left).mpr hx.right.le
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· -- log is integrable on [[0, t]]
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simp [Set.mem_uIcc, ht]
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@ -6,6 +6,7 @@ import Mathlib.MeasureTheory.Measure.Restrict
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open scoped Interval Topology
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open Real Filter MeasureTheory intervalIntegral
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-- The following theorem was suggested by Gareth Ma on Zulip
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lemma logsinBound : ∀ x ∈ (Set.Icc 0 1), ‖(log ∘ sin) x‖ ≤ ‖log ((π / 2)⁻¹ * x)‖ := by
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@ -226,6 +227,62 @@ lemma intervalIntegrable_log_cos : IntervalIntegrable (log ∘ cos) volume 0 (π
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rwa [← this]
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theorem logInt
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{t : ℝ}
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(ht : 0 < t) :
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∫ x in (0 : ℝ)..t, log x = t * log t - t := by
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rw [← integral_add_adjacent_intervals (b := 1)]
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trans (-1) + (t * log t - t + 1)
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· congr
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· -- ∫ x in 0..1, log x = -1, same proof as before
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rw [integral_eq_sub_of_hasDerivAt_of_tendsto (f := fun x ↦ x * log x - x) (fa := 0) (fb := -1)]
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· simp
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· simp
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· intro x hx
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norm_num at hx
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convert (hasDerivAt_mul_log hx.left.ne.symm).sub (hasDerivAt_id x) using 1
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norm_num
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· rw [← neg_neg log]
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apply IntervalIntegrable.neg
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apply intervalIntegrable_deriv_of_nonneg (g := fun x ↦ -(x * log x - x))
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· exact (continuous_mul_log.continuousOn.sub continuous_id.continuousOn).neg
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· intro x hx
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norm_num at hx
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convert ((hasDerivAt_mul_log hx.left.ne.symm).sub (hasDerivAt_id x)).neg using 1
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norm_num
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· intro x hx
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norm_num at hx
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rw [Pi.neg_apply, Left.nonneg_neg_iff]
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exact (log_nonpos_iff hx.left).mpr hx.right.le
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· have := tendsto_log_mul_rpow_nhds_zero zero_lt_one
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simp_rw [rpow_one, mul_comm] at this
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-- tendsto_nhdsWithin_of_tendsto_nhds should be under Tendsto namespace
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convert this.sub (tendsto_nhdsWithin_of_tendsto_nhds tendsto_id)
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norm_num
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· rw [(by simp : -1 = 1 * log 1 - 1)]
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apply tendsto_nhdsWithin_of_tendsto_nhds
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exact (continuousAt_id.mul (continuousAt_log one_ne_zero)).sub continuousAt_id
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· -- ∫ x in 1..t, log x = t * log t + 1, just use integral_log_of_pos
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rw [integral_log_of_pos zero_lt_one ht]
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norm_num
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· abel
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· -- log is integrable on [[0, 1]]
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rw [← neg_neg log]
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apply IntervalIntegrable.neg
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apply intervalIntegrable_deriv_of_nonneg (g := fun x ↦ -(x * log x - x))
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· exact (continuous_mul_log.continuousOn.sub continuous_id.continuousOn).neg
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· intro x hx
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norm_num at hx
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convert ((hasDerivAt_mul_log hx.left.ne.symm).sub (hasDerivAt_id x)).neg using 1
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norm_num
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· intro x hx
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norm_num at hx
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rw [Pi.neg_apply, Left.nonneg_neg_iff]
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exact (log_nonpos_iff hx.left).mpr hx.right.le
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· -- log is integrable on [[0, t]]
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simp [Set.mem_uIcc, ht]
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lemma integral_log_sin : ∫ (x : ℝ) in (0)..(π / 2), log (sin x) = -log 2 * π/2 := by
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have t₀ {x : ℝ} : sin (2 * x) = 2 * sin x * cos x := sin_two_mul x
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@ -276,3 +333,56 @@ lemma integral_log_sin : ∫ (x : ℝ) in (0)..(π / 2), log (sin x) = -log 2 *
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simp
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-- -- IntervalIntegrable (fun x => log (cos x)) volume 0 (π / 2)
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exact intervalIntegrable_log_cos
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lemma int₁₁ : ∫ (x : ℝ) in (0)..π, log (4 * sin x ^ 2) = 0 := by
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sorry
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lemma int₁ :
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∫ x in (0)..(2 * π), log ‖circleMap 0 1 x - 1‖ = 0 := by
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have {x : ℝ} : log ‖circleMap 0 1 x - 1‖ = log (4 * sin (x / 2) ^ 2) / 2 := by
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dsimp [Complex.abs]
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rw [log_sqrt (Complex.normSq_nonneg (circleMap 0 1 x - 1))]
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congr
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calc Complex.normSq (circleMap 0 1 x - 1)
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_ = (cos x - 1) * (cos x - 1) + sin x * sin x := by
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dsimp [circleMap]
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rw [Complex.normSq_apply]
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simp
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_ = sin x ^ 2 + cos x ^ 2 + 1 - 2 * cos x := by
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ring
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_ = 2 - 2 * cos x := by
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rw [sin_sq_add_cos_sq]
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norm_num
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_ = 2 - 2 * cos (2 * (x / 2)) := by
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rw [← mul_div_assoc]
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congr; norm_num
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_ = 4 - 4 * Real.cos (x / 2) ^ 2 := by
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rw [cos_two_mul]
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ring
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_ = 4 * sin (x / 2) ^ 2 := by
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nth_rw 1 [← mul_one 4, ← sin_sq_add_cos_sq (x / 2)]
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ring
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simp_rw [this]
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simp
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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
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have : 1 = 2 * (2 : ℝ)⁻¹ := by exact Eq.symm (mul_inv_cancel_of_invertible 2)
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nth_rw 1 [← one_mul (∫ (x : ℝ) in (0)..2 * π, log (4 * sin (x / 2) ^ 2))]
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rw [← mul_inv_cancel_of_invertible 2, mul_assoc]
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let f := fun y ↦ log (4 * sin y ^ 2)
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have {x : ℝ} : log (4 * sin (x / 2) ^ 2) = f (x / 2) := by simp
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conv =>
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left
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right
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right
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arg 1
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intro x
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rw [this]
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rw [intervalIntegral.inv_mul_integral_comp_div 2]
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simp
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rw [this]
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simp
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exact int₁₁
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