| import OAI.NumberTheory.Jacobsthal.Analysis.TerminalForceIntegral |
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| namespace OAI |
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| namespace Erdos970 |
| open scoped _root_.Erdos970 |
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| section |
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| open _root_.Set _root_.MeasureTheory |
| namespace ErdosOmissionTail |
| open ErdosContinuousOmission ErdosContinuousBoundary |
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| noncomputable def windowKernel (S a u c : ℝ) : ℝ := |
| ((S+a+u+2*c)^3-(S+2*a)^3)/(12*(max 1 a)*(max 1 u)*(max 1 c)^2) |
| noncomputable def windowMiddle (S a u : ℝ) : ℝ := |
| ∫ c : ℝ in 1..baseCutoff u,windowKernel S a u c |
| noncomputable def windowOuter (S a : ℝ) : ℝ := |
| ∫ u : ℝ in 1..baseCutoff a,windowMiddle S a u |
| noncomputable def windowMass (S b : ℝ) : ℝ := |
| ∫ a : ℝ in 1..baseCutoff b,windowOuter S a |
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|
| theorem windowKernel_continuous (S : ℝ) : |
| Continuous (fun q : (ℝ×ℝ)×ℝ => windowKernel S q.1.1 q.1.2 q.2) := by |
| have hn : Continuous (fun q : (ℝ×ℝ)×ℝ => (S+q.1.1+q.1.2+2*q.2)^3-(S+2*q.1.1)^3) := by fun_prop |
| have hd : Continuous (fun q : (ℝ×ℝ)×ℝ => 12*(max 1 q.1.1)*(max 1 q.1.2)*(max 1 q.2)^2) := by fun_prop |
| apply hn.div hd |
| intro q |
| have h1 : 0 < max (1:ℝ) q.1.1 := (by norm_num : (0:ℝ)<1).trans_le (le_max_left _ _) |
| have h2 : 0 < max (1:ℝ) q.1.2 := (by norm_num : (0:ℝ)<1).trans_le (le_max_left _ _) |
| have h3 : 0 < max (1:ℝ) q.2 := (by norm_num : (0:ℝ)<1).trans_le (le_max_left _ _) |
| positivity |
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| theorem windowMiddle_continuous (S : ℝ) : Continuous (fun p : ℝ×ℝ => windowMiddle S p.1 p.2) := by |
| exact continuous_variable_upper_integral (fun p c => windowKernel S p.1 p.2 c) |
| (windowKernel_continuous S) (fun p => baseCutoff p.2) |
| (continuous_const.max (continuous_const.min continuous_snd)) |
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| theorem windowOuter_continuous (S : ℝ) : Continuous (windowOuter S) := by |
| have hp : Continuous (fun q : (ℝ×ℝ)×ℝ => (q.1.1,q.2)) := by fun_prop |
| have hF : Continuous (fun q : (ℝ×ℝ)×ℝ => windowMiddle S q.1.1 q.2) := |
| (windowMiddle_continuous S).comp hp |
| have hcap : Continuous (fun p : ℝ×ℝ => baseCutoff p.1) := |
| continuous_const.max (continuous_const.min continuous_fst) |
| have hh : Continuous (fun p : ℝ×ℝ => ∫ u : ℝ in 1..baseCutoff p.1,windowMiddle S p.1 u) := |
| continuous_variable_upper_integral (fun p u => windowMiddle S p.1 u) hF (fun p => baseCutoff p.1) hcap |
| have hmap : Continuous (fun a : ℝ => (a,(0:ℝ))) := by fun_prop |
| change Continuous (fun a : ℝ => ∫ u : ℝ in 1..baseCutoff a,windowMiddle S a u) |
| simpa only [Function.comp_def] using! hh.comp hmap |
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| theorem windowKernel_literal {S a u c : ℝ} (ha : 1 ≤ a) (hu : 1 ≤ u) (hc : 1 ≤ c) : |
| windowKernel S a u c=lastWindowCoefficient S a u c/(a*u*c) := by |
| rw [windowKernel,max_eq_right ha,max_eq_right hu,max_eq_right hc,lastWindowCoefficient] |
| ring |
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| theorem scalarTailMass_zero_eq_windowMass {S b : ℝ} (hS : 0 ≤ S) (hb1 : 1 ≤ b) (hb2 : b ≤ 2) : |
| scalarTailMass 0 S b=windowMass S b := by |
| rw [scalarTailMass_zero_triple hS hb1 hb2,windowMass,baseCutoff_on hb1 hb2] |
| apply intervalIntegral.integral_congr |
| intro a ha |
| rw [uIcc_of_le hb1] at ha |
| dsimp only |
| rw [windowOuter,baseCutoff_on ha.1 (ha.2.trans hb2)] |
| apply intervalIntegral.integral_congr |
| intro u hu |
| rw [uIcc_of_le ha.1] at hu |
| dsimp only |
| rw [windowMiddle,baseCutoff_on hu.1 (hu.2.trans (ha.2.trans hb2))] |
| apply intervalIntegral.integral_congr |
| intro c hc |
| rw [uIcc_of_le hu.1] at hc |
| exact (windowKernel_literal ha.1 hu.1 hc.1).symm |
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| end ErdosOmissionTail |
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| end |
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| end Erdos970 |
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| end OAI |
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