| import OAI.NumberTheory.Jacobsthal.Estimates.ScalarTailRecurrence |
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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 ErdosOmissionBindings |
| open NumberTheoryLean.FinitePathGeometry |
|
|
| theorem terminalBlock_mask (S y : ℝ) {b : ℝ} (_hb1 : 1 ≤ b) (hb2 : b ≤ 2) : |
| terminalBlock (2*y+2-S) b= |
| ∫ a in Icc (1:ℝ) b,if S/2+a ≤ y then |
| gate omissionForce .even (2*y+2-S-a) a/(max 1 a) else 0 := by |
| have hset : Ioc (1:ℝ) (upperCutoff .odd (2*y+2-S) b)= |
| Ioc (1:ℝ) b ∩ {a : ℝ | S/2+a ≤ y} := by |
| ext a |
| constructor |
| · intro ha |
| obtain ⟨ha1,ha2,hab,hgate⟩ := (mem_upperCutoff .odd (2*y+2-S) b a).mp ha |
| have hh := (strong_gate_below_two ha2).mp hgate |
| exact ⟨⟨ha1,hab⟩,by change S/2+a ≤ y; linarith⟩ |
| · rintro ⟨⟨ha1,hab⟩,hy⟩ |
| apply (mem_upperCutoff .odd (2*y+2-S) b a).mpr |
| refine ⟨ha1,hab.trans hb2,hab,?_⟩ |
| apply (strong_gate_below_two (hab.trans hb2)).mpr |
| change S/2+a ≤ y at hy |
| linarith |
| have hm : MeasurableSet {a : ℝ | S/2+a ≤ y} := |
| measurableSet_le (measurable_const.add measurable_id) measurable_const |
| change (∫ a : ℝ in 1..upperCutoff .odd (2*y+2-S) b, |
| gate omissionForce .even (2*y+2-S-a) a/(max 1 a))=_ |
| rw [intervalIntegral.integral_of_le (upperCutoff_bounds .odd (2*y+2-S) b).1, |
| hset,← setIntegral_indicator hm,← integral_Icc_eq_integral_Ioc] |
| rfl |
|
|
| noncomputable def terminalOuterIntegrand (S : ℝ) (p : ℝ×ℝ) : ℝ := |
| if S/2+p.2 ≤ p.1 then |
| 2*p.1^2*gate omissionForce .even (2*p.1+2-S-p.2) p.2/(max 1 p.2) else 0 |
|
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| theorem terminalOuterIntegrand_measurable (S : ℝ) : Measurable (terminalOuterIntegrand S) := by |
| have hc0 : Continuous (fun p : ℝ×ℝ => gate omissionForce .even p.1 p.2) := |
| gate_continuous omissionForce_continuous .even |
| have hp : Continuous (fun p : ℝ×ℝ => (2*p.1+2-S-p.2,p.2)) := by fun_prop |
| have hc : Continuous (fun p : ℝ×ℝ => gate omissionForce .even (2*p.1+2-S-p.2) p.2) := hc0.comp hp |
| exact Measurable.ite (measurableSet_le (measurable_const.add measurable_snd) measurable_fst) |
| (((measurable_const.mul (measurable_fst.pow_const 2)).mul hc.measurable).div |
| (measurable_const.max measurable_snd)) measurable_const |
|
|
| theorem terminalOuterIntegrand_bound (S a : ℝ) {y Y : ℝ} (hy0 : 0 ≤ y) (hyY : y ≤ Y) : |
| |terminalOuterIntegrand S (y,a)| ≤ 2*Y^2 := by |
| unfold terminalOuterIntegrand |
| split_ifs |
| · have hp := gate_nonnegative omissionForce_nonnegative .even (2*y+2-S-a) a |
| have hu := gateIterate_force_le_one 1 .even (2*y+2-S-a) a |
| change gate omissionForce .even (2*y+2-S-a) a ≤ 1 at hu |
| have hn : 0 ≤ 2*y^2*gate omissionForce .even (2*y+2-S-a) a := by positivity |
| have hm := mul_le_mul_of_nonneg_left hu (mul_nonneg (by norm_num : (0:ℝ)≤2) (sq_nonneg y)) |
| rw [abs_of_nonneg (div_nonneg hn (zero_le_one.trans (le_max_left _ _)))] |
| apply (div_le_self hn (le_max_left (1:ℝ) a)).trans |
| nlinarith |
| · simpa only [abs_zero] using mul_nonneg (by norm_num : (0:ℝ)≤2) (sq_nonneg Y) |
|
|
| theorem scalarTailMass_zero_outer {S b : ℝ} (hS : 0 ≤ S) (hb1 : 1 ≤ b) (hb2 : b ≤ 2) : |
| scalarTailMass 0 S b= |
| ∫ a : ℝ in 1..b,(2/a)*(∫ y : ℝ in S/2+a..S/2+4, |
| y^2*gate omissionForce .even (2*y+2-S-a) a) := by |
| let Y : ℝ := S/2+4 |
| have hY : 0 ≤ Y := by dsimp [Y]; linarith |
| have hl (y : ℝ) : scalarTailIntegrand 0 S b y=∫ a in Icc (1:ℝ) b,terminalOuterIntegrand S (y,a) := by |
| change 2*y^2*terminalBlock (2*y+2-S) b=_ |
| rw [terminalBlock_mask S y hb1 hb2,← integral_const_mul] |
| apply integral_congr_ae |
| filter_upwards with a |
| dsimp only [terminalOuterIntegrand] |
| split_ifs <;> ring |
| have hs := bounded_rectangle_swap (terminalOuterIntegrand_measurable S) 0 Y 1 b (2*Y^2) |
| (fun y hy a _ha => terminalOuterIntegrand_bound S a hy.1 hy.2) |
| have hr (a : ℝ) (ha : a ∈ Icc (1:ℝ) b) : |
| (∫ y in Icc (0:ℝ) Y,terminalOuterIntegrand S (y,a))= |
| (2/a)*(∫ y : ℝ in S/2+a..Y,y^2*gate omissionForce .even (2*y+2-S-a) a) := by |
| have ha2 : a ≤ 2 := ha.2.trans hb2 |
| have hlo : 0 ≤ S/2+a := by linarith [ha.1] |
| have hhi : S/2+a ≤ Y := by dsimp [Y]; linarith |
| have hset : Icc (0:ℝ) Y ∩ Ici (S/2+a)=Icc (S/2+a) Y := by |
| ext y |
| constructor |
| · rintro ⟨hy,hylo⟩; exact ⟨hylo,hy.2⟩ |
| · intro hy; exact ⟨⟨hlo.trans hy.1,hy.2⟩,hy.1⟩ |
| have he : (fun y : ℝ => terminalOuterIntegrand S (y,a))= |
| (Ici (S/2+a)).indicator (fun y => (2/a)*(y^2*gate omissionForce .even (2*y+2-S-a) a)) := by |
| funext y |
| unfold terminalOuterIntegrand |
| by_cases h : S/2+a ≤ y |
| · rw [ite_eq_left h,indicator_of_mem (show y ∈ Ici (S/2+a) from h),max_eq_right ha.1] |
| ring |
| · rw [ite_eq_right h,indicator_of_notMem (show y ∉ Ici (S/2+a) from h)] |
| rw [he,setIntegral_indicator measurableSet_Ici,hset,integral_const_mul, |
| integral_Icc_eq_integral_Ioc,← intervalIntegral.integral_of_le hhi] |
| rw [scalarTailMass_bounded 0 S b Y hY (by dsimp [Y]; norm_num), |
| intervalIntegral.integral_of_le hY,← integral_Icc_eq_integral_Ioc] |
| simp_rw [hl] |
| rw [hs,intervalIntegral.integral_of_le hb1,← integral_Icc_eq_integral_Ioc] |
| exact setIntegral_congr_fun measurableSet_Icc hr |
|
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| end ErdosOmissionTail |
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| end |
|
|
| section |
|
|
| open _root_.Set _root_.MeasureTheory |
| namespace ErdosOmissionTail |
| open ErdosContinuousOmission ErdosContinuousBoundary ErdosOmissionBindings |
| open NumberTheoryLean.FinitePathGeometry |
|
|
| theorem evenGate_weighted_swap (S a l Y : ℝ) (ha1 : 1 ≤ a) (ha2 : a ≤ 2) |
| (hl : 0 ≤ l) (hlY : l ≤ Y) : |
| (∫ y : ℝ in l..Y,y^2*gate omissionForce .even (2*y+2-S-a) a)= |
| ∫ u : ℝ in 1..a,(∫ y : ℝ in l..Y,y^2*omissionForce .odd (2*y+2-S-a-u) u)/u := by |
| let G : ℝ×ℝ → ℝ := fun p => p.1^2*omissionForce .odd (2*p.1+2-S-a-p.2) p.2/(max 1 p.2) |
| have hpmap : Continuous (fun p : ℝ×ℝ => (2*p.1+2-S-a-p.2,p.2)) := by fun_prop |
| have hc : Continuous (fun p : ℝ×ℝ => omissionForce .odd (2*p.1+2-S-a-p.2) p.2) := |
| (omissionForce_continuous .odd).comp hpmap |
| have hG : Continuous G := by |
| apply ((continuous_fst.pow 2).mul hc).div (continuous_const.max continuous_snd) |
| intro p |
| exact ne_of_gt ((by norm_num : (0:ℝ)<1).trans_le (le_max_left _ _)) |
| have hbnd : ∀ y ∈ Icc l Y,∀ u ∈ Icc (1:ℝ) a,|G (y,u)| ≤ Y^2 := by |
| intro y hy u _hu |
| have hy0 : 0 ≤ y := hl.trans hy.1 |
| have hp := omissionForce_nonnegative .odd (2*y+2-S-a-u) u |
| have hu := omissionForce_le_one .odd (2*y+2-S-a-u) u |
| have hn : 0 ≤ y^2*omissionForce .odd (2*y+2-S-a-u) u := by positivity |
| have hh := mul_le_mul_of_nonneg_left hu (sq_nonneg y) |
| dsimp only [G] |
| rw [abs_of_nonneg (div_nonneg hn (zero_le_one.trans (le_max_left _ _)))] |
| apply (div_le_self hn (le_max_left (1:ℝ) u)).trans |
| nlinarith [hy.2] |
| have hs := bounded_rectangle_swap hG.measurable l Y 1 a (Y^2) hbnd |
| have hs' : (∫ y : ℝ in l..Y,∫ u : ℝ in 1..a,G (y,u))= |
| ∫ u : ℝ in 1..a,∫ y : ℝ in l..Y,G (y,u) := by |
| simpa only [integral_Icc_eq_integral_Ioc,← intervalIntegral.integral_of_le hlY, |
| ← intervalIntegral.integral_of_le ha1] using hs |
| have hl' : (∫ y : ℝ in l..Y,y^2*gate omissionForce .even (2*y+2-S-a) a)= |
| ∫ y : ℝ in l..Y,∫ u : ℝ in 1..a,G (y,u) := by |
| apply intervalIntegral.integral_congr |
| intro y _hy |
| dsimp only |
| rw [gate,show upperCutoff .even (2*y+2-S-a) a=a from baseCutoff_on ha1 ha2, |
| ← intervalIntegral.integral_const_mul] |
| apply intervalIntegral.integral_congr |
| intro u _hu |
| dsimp only [G,Side.flip] |
| ring |
| rw [hl',hs'] |
| apply intervalIntegral.integral_congr |
| intro u hu |
| rw [uIcc_of_le ha1] at hu |
| dsimp only |
| rw [← intervalIntegral.integral_div] |
| apply intervalIntegral.integral_congr |
| intro y _hy |
| dsimp only [G] |
| rw [max_eq_right hu.1] |
|
|
| theorem scalarTailMass_zero_triple {S b : ℝ} (hS : 0 ≤ S) (hb1 : 1 ≤ b) (hb2 : b ≤ 2) : |
| scalarTailMass 0 S b= |
| ∫ a : ℝ in 1..b,∫ u : ℝ in 1..a,∫ c : ℝ in 1..u, |
| lastWindowCoefficient S a u c/(a*u*c) := by |
| rw [scalarTailMass_zero_outer hS hb1 hb2] |
| apply intervalIntegral.integral_congr |
| intro a ha |
| rw [uIcc_of_le hb1] at ha |
| have ha2 : a ≤ 2 := ha.2.trans hb2 |
| have hl : 0 ≤ S/2+a := by linarith [ha.1] |
| have hlY : S/2+a ≤ S/2+4 := by linarith |
| dsimp only |
| rw [evenGate_weighted_swap S a (S/2+a) (S/2+4) ha.1 ha2 hl hlY, |
| ← intervalIntegral.integral_const_mul] |
| apply intervalIntegral.integral_congr |
| intro u hu |
| rw [uIcc_of_le ha.1] at hu |
| have hu2 : u ≤ 2 := hu.2.trans ha2 |
| have hw := integrated_force_window (S+a+u) (S/2+a) (S/2+4) u 0 hl hu.1 hu2 |
| (by linarith [hu.1]) (by linarith) |
| simp only [sub_zero,zero_mul] at hw |
| have hgap (y : ℝ) : 2*y+2-S-a-u=2*y+2-(S+a+u) := by ring |
| dsimp only |
| simp_rw [hgap] |
| rw [hw,← intervalIntegral.integral_div,← intervalIntegral.integral_const_mul] |
| apply intervalIntegral.integral_congr |
| intro c hc |
| rw [uIcc_of_le hu.1] at hc |
| have ha0 : a ≠ 0 := by linarith [ha.1] |
| have hu0 : u ≠ 0 := by linarith [hu.1] |
| have hc0 : c ≠ 0 := by linarith [hc.1] |
| dsimp only [lastWindowCoefficient] |
| field_simp |
| ring |
|
|
| end ErdosOmissionTail |
|
|
| end |
|
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| end Erdos970 |
|
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| end OAI |
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