File size: 4,182 Bytes
f89a930 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 | import OAI.NumberTheory.Jacobsthal.Probability.ThirdInnerKernel
namespace OAI
namespace Erdos970
open scoped _root_.Erdos970
section
open _root_.Set _root_.Erdos970.Set _root_.MeasureTheory _root_.Erdos970.MeasureTheory
namespace ErdosOmissionBindings
open ErdosContinuousOmission ErdosContinuousBoundary ErdosBoundaryLoss
noncomputable def thirdLowerIntegral (p : ℝ×ℝ) : ℝ :=
∫ t : ℝ in 1..baseCutoff p.2,clampedThird p.1 p.2 t
noncomputable def thirdUpperIntegral (p : ℝ×ℝ) : ℝ :=
∫ x : ℝ in baseCutoff p.1..2,clampedThird x p.1 p.2
theorem thirdLowerIntegral_continuous : Continuous thirdLowerIntegral := by
let F : (ℝ×ℝ) → ℝ → ℝ := fun p t => clampedThird p.1 p.2 t
have hF : Continuous F.uncurry := by
change Continuous ((fun q : ℝ×ℝ×ℝ => clampedThird q.1 q.2.1 q.2.2) ∘
(fun p : (ℝ×ℝ)×ℝ => (p.1.1,p.1.2,p.2)))
exact clampedThird_continuous.comp
(continuous_fst.fst.prodMk (continuous_fst.snd.prodMk continuous_snd))
exact continuous_variable_upper_integral F hF (fun p => baseCutoff p.2)
(baseCutoff_continuous.comp continuous_snd)
theorem thirdUpperIntegral_continuous : Continuous thirdUpperIntegral := by
let F : (ℝ×ℝ) → ℝ → ℝ := fun p x => clampedThird x p.1 p.2
have hF : Continuous F.uncurry := by
change Continuous ((fun q : ℝ×ℝ×ℝ => clampedThird q.1 q.2.1 q.2.2) ∘
(fun p : (ℝ×ℝ)×ℝ => (p.2,p.1.1,p.1.2)))
exact clampedThird_continuous.comp
(continuous_snd.prodMk (continuous_fst.fst.prodMk continuous_fst.snd))
have h2 := continuous_variable_upper_integral F hF (fun _ => (2:ℝ)) continuous_const
have hc := continuous_variable_upper_integral F hF (fun p => baseCutoff p.1)
(baseCutoff_continuous.comp continuous_fst)
have he (p : ℝ×ℝ) : (∫ x : ℝ in 1..2,F p x)-(∫ x : ℝ in 1..baseCutoff p.1,F p x)=thirdUpperIntegral p := by
have hf : Continuous (F p) := by
change Continuous (F.uncurry ∘ (fun x : ℝ => (p,x)))
exact hF.comp (continuous_const.prodMk continuous_id)
have hi := intervalIntegral.integral_add_adjacent_intervals
(hf.intervalIntegrable (μ := volume) 1 (baseCutoff p.1))
(hf.intervalIntegrable (μ := volume) (baseCutoff p.1) 2)
change (∫ x in 1..2,F p x)-(∫ x in 1..baseCutoff p.1,F p x)=∫ x in baseCutoff p.1..2,F p x
linarith
exact (h2.sub hc).congr he
theorem thirdLowerIntegral_bound (p : ℝ×ℝ) : |thirdLowerIntegral p| ≤ 64 := by
have h := intervalIntegral.norm_integral_le_of_norm_le_const (a:=1) (b:=baseCutoff p.2)
(C:=64) (f:=fun t => clampedThird p.1 p.2 t)
(fun t _ => by simpa only [Real.norm_eq_abs] using clampedThird_bound p.1 p.2 t)
have hc := baseCutoff_bounds p.2
change |thirdLowerIntegral p| ≤ 64
rw [Real.norm_eq_abs,abs_of_nonneg (by linarith : 0 ≤ baseCutoff p.2-1)] at h
exact h.trans (by nlinarith)
theorem thirdUpperIntegral_bound (p : ℝ×ℝ) : |thirdUpperIntegral p| ≤ 64 := by
have h := intervalIntegral.norm_integral_le_of_norm_le_const (a:=baseCutoff p.1) (b:=2)
(C:=64) (f:=fun x => clampedThird x p.1 p.2)
(fun x _ => by simpa only [Real.norm_eq_abs] using clampedThird_bound x p.1 p.2)
have hc := baseCutoff_bounds p.1
change |thirdUpperIntegral p| ≤ 64
rw [Real.norm_eq_abs,abs_of_nonneg (by linarith : 0 ≤ 2-baseCutoff p.1)] at h
exact h.trans (by nlinarith)
theorem thirdLowerIntegral_eq {x u : ℝ} (hx : x ∈ Icc (1:ℝ) 2) (hu : u ∈ Icc (1:ℝ) 2) :
thirdLowerIntegral (x,u)=∫ t : ℝ in 1..u,loss3Kernel t u x := by
rw [thirdLowerIntegral,baseCutoff_on hu.1 hu.2]
apply intervalIntegral.integral_congr
intro t ht
rw [uIcc_of_le hu.1] at ht
exact clampedThird_eq hx hu ⟨ht.1,ht.2.trans hu.2⟩
theorem thirdUpperIntegral_eq {u t : ℝ} (hu : u ∈ Icc (1:ℝ) 2) (ht : t ∈ Icc (1:ℝ) 2) :
thirdUpperIntegral (u,t)=∫ x : ℝ in u..2,loss3Kernel t u x := by
rw [thirdUpperIntegral,baseCutoff_on hu.1 hu.2]
apply intervalIntegral.integral_congr
intro x hx
rw [uIcc_of_le hu.2] at hx
exact clampedThird_eq ⟨hu.1.trans hx.1,hx.2⟩ hu ht
end ErdosOmissionBindings
end
end Erdos970
end OAI
|