Documentation

SpherePacking.MagicFunction.a.IntegralEstimates.I5

Constructing Upper-Bounds for I₅ #

The purpose of this file is to construct bounds on the integral I₅ that is part of the definition of the function a. We follow the proof of Proposition 7.8 in the blueprint.

TODO: #

We begin by performing changes of variables. We use Ioc intervals everywhere because of the way intervalIntegral is defined.

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def MagicFunction.a.IntegralEstimates.I₅.f :
ℝ → ℝ
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    def MagicFunction.a.IntegralEstimates.I₅.f' :
    ℝ → ℝ
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      def MagicFunction.a.IntegralEstimates.I₅.g :
      ℝ → ℝ → ℂ
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        theorem MagicFunction.a.IntegralEstimates.I₅.aux_measurable :
        MeasurableSet (Set.Ioc 0 1)
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        theorem MagicFunction.a.IntegralEstimates.I₅.aux_hasDeriv (x : ℝ) (hx : x ∈ Set.Ioc 0 1) :
        HasDerivWithinAt f (f' x) (Set.Ioc 0 1) x
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        theorem MagicFunction.a.IntegralEstimates.I₅.aux_injOn :
        Set.InjOn f (Set.Ioc 0 1)
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        theorem MagicFunction.a.IntegralEstimates.I₅.Changing_Domain_of_Integration (r : ℝ) :
        ∫ (s : ℝ) in Set.Ici 1, g r s = ∫ (s : ℝ) in f '' Set.Ioc 0 1, g r s
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        theorem MagicFunction.a.IntegralEstimates.I₅.Changing_Variables (r : ℝ) :
        ∫ (s : ℝ) in f '' Set.Ioc 0 1, g r s = ∫ (t : ℝ) in Set.Ioc 0 1, |f' t| • g r (f t)
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        theorem MagicFunction.a.IntegralEstimates.I₅.Writing_as_intervalIntegral (r : ℝ) :
        ∫ (t : ℝ) in Set.Ioc 0 1, |f' t| • g r (f t) = ∫ (t : ℝ) in 0 ..1, |f' t| • g r (f t)
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        theorem MagicFunction.a.IntegralEstimates.I₅.Reconciling_Change_of_Variables (r : ℝ) :
        RealIntegrals.I₅' r = -2 * ∫ (t : ℝ) in Set.Ioc 0 1, |f' t| • g r (f t)
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        theorem MagicFunction.a.IntegralEstimates.I₅.Complete_Change_of_Variables (r : ℝ) :
        RealIntegrals.I₅' r = -2 * ∫ (s : ℝ) in Set.Ici 1, g r s
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        theorem MagicFunction.a.IntegralEstimates.I₅.I₅'_bounding_aux_1 (r x : ℝ) :
        x ∈ Set.Ici 1 → ‖g r x‖ ≤ ‖φ₀'' (Complex.I * ↑x)‖ * Real.exp (-Real.pi * r / x)
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        theorem MagicFunction.a.IntegralEstimates.I₅.I₅'_bounding_aux_2 (r : ℝ) :
        ∃ C₀ > 0, ∀ x ∈ Set.Ici 1, ‖g r x‖ ≤ C₀ * Real.exp (-2 * Real.pi * x) * Real.exp (-Real.pi * r / x)
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        theorem MagicFunction.a.IntegralEstimates.I₅.Bound_integrableOn (r C₀ : ℝ) (hC₀_pos : C₀ > 0) (hC₀ : ∀ x ∈ Set.Ici 1, ‖g r x‖ ≤ C₀ * Real.exp (-2 * Real.pi * x) * Real.exp (-Real.pi * r / x)) :
        MeasureTheory.IntegrableOn (fun (s : ℝ) => C₀ * Real.exp (-2 * Real.pi * s) * Real.exp (-Real.pi * r / s)) (Set.Ici 1) MeasureTheory.volume
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        theorem MagicFunction.a.IntegralEstimates.I₅.I₅'_bounding_1_aux_3 (r : ℝ) :
        ∃ C₀ > 0, ∫ (s : ℝ) in Set.Ici 1, ‖g r s‖ ≤ ∫ (s : ℝ) in Set.Ici 1, C₀ * Real.exp (-2 * Real.pi * s) * Real.exp (-Real.pi * r / s)
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        theorem MagicFunction.a.IntegralEstimates.I₅.I₅'_bounding (r : ℝ) :
        ∃ C₀ > 0, ‖RealIntegrals.I₅' r‖ ≤ 2 * ∫ (s : ℝ) in Set.Ici 1, C₀ * Real.exp (-2 * Real.pi * s) * Real.exp (-Real.pi * r / s)