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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 93 94 | import Mathlib.MeasureTheory.Group.Integral
import OAI.NumberTheory.Jacobsthal.Estimates.InvariantFiniteness
namespace OAI
namespace Erdos970
section
open Set MeasureTheory
namespace Erdos970Dependency.DeficitIntegralValues
open NumberTheoryLean.LinearSieveFunctions NumberTheoryLean.BuchstabBridge
open NumberTheoryLean.NormalizedDeficits NumberTheoryLean.DeficitFutureIntegrals
theorem shift_indicator (f : ℝ → ℝ) (a d : ℝ) :
(fun s ↦ (Ioi a).indicator f (s - d)) = (Ioi (a + d)).indicator (fun s ↦ f (s - d)) := by
funext s
by_cases hs : a + d < s
· rw [indicator_of_mem (show s ∈ Ioi (a + d) from hs),
indicator_of_mem (show s - d ∈ Ioi a by change a < s - d; linarith)]
· rw [indicator_of_notMem (show s ∉ Ioi (a + d) from hs),
indicator_of_notMem (show s - d ∉ Ioi a by change ¬a < s - d; linarith)]
theorem integrableOn_shift_Ioi {f : ℝ → ℝ} {a : ℝ} (hf : IntegrableOn f (Ioi a)) (d : ℝ) :
IntegrableOn (fun s ↦ f (s - d)) (Ioi (a + d)) := by
have h := ((integrable_indicator_iff measurableSet_Ioi).mpr hf).comp_sub_right d
rw [shift_indicator] at h
exact (integrable_indicator_iff measurableSet_Ioi).mp h
theorem integral_shift_Ioi (f : ℝ → ℝ) (a d : ℝ) :
(∫ s in Ioi (a + d), f (s - d)) = ∫ s in Ioi a, f s := by
have h := integral_sub_right_eq_self (μ := (volume : Measure ℝ)) ((Ioi a).indicator f) d
rw [shift_indicator, integral_indicator measurableSet_Ioi, integral_indicator measurableSet_Ioi] at h
exact h
theorem upper_integral_one : IntegrableOn upperDeficit (Ioi (1 : ℝ)) ∧
(∫ s in Ioi (1 : ℝ), upperDeficit s) = 2 := by
have h := lower_future_integral (s := 2) (by norm_num)
norm_num only [show (2 : ℝ) - 1 = 1 by norm_num] at h
refine ⟨h.1, ?_⟩
rw [← h.2]
norm_num [lowerDeficit, f_initial sieveA (s := 2) (by norm_num)]
theorem lower_integral_zero : IntegrableOn lowerDeficit (Ioi (0 : ℝ)) ∧
(∫ s in Ioi (0 : ℝ), lowerDeficit s) = sieveA - 1 := by
have h := upper_future_integral (s := 1) (by norm_num)
norm_num only [show (1 : ℝ) - 1 = 0 by norm_num, one_mul] at h
refine ⟨h.1, ?_⟩
rw [← h.2]
norm_num [upperDeficit, F_initial sieveA (s := 1) (by norm_num) (by norm_num)]
theorem lower_integral_two : IntegrableOn lowerDeficit (Ioi (2 : ℝ)) ∧
(∫ s in Ioi (2 : ℝ), lowerDeficit s) = sieveA - 3 := by
have h := upper_future_integral (s := 3) (by norm_num)
norm_num only [show (3 : ℝ) - 1 = 2 by norm_num] at h
refine ⟨h.1, ?_⟩
rw [← h.2]
simp only [upperDeficit, F_initial sieveA (s := 3) (by norm_num) (by norm_num)]
ring
theorem even_majorant_integral :
IntegrableOn (fun s ↦ lowerDeficit s + upperDeficit (s - 1)) (Ici (2 : ℝ)) ∧
(∫ s in Ici (2 : ℝ), lowerDeficit s + upperDeficit (s - 1)) = sieveA - 1 := by
have ha := lower_integral_two.1
have hc : IntegrableOn (fun s ↦ upperDeficit (s - 1)) (Ioi (2 : ℝ)) := by
simpa only [show (1 : ℝ) + 1 = 2 by norm_num] using integrableOn_shift_Ioi upper_integral_one.1 1
refine ⟨Iff.mpr integrableOn_Ici_iff_integrableOn_Ioi (ha.add hc), ?_⟩
rw [integral_Ici_eq_integral_Ioi, integral_add ha hc, lower_integral_two.2]
have h := integral_shift_Ioi upperDeficit 1 1
norm_num only [show (1 : ℝ) + 1 = 2 by norm_num] at h
rw [h, upper_integral_one.2]
ring
theorem odd_majorant_integral :
IntegrableOn (fun s ↦ upperDeficit s + lowerDeficit (s - 1)) (Ici (1 : ℝ)) ∧
(∫ s in Ici (1 : ℝ), upperDeficit s + lowerDeficit (s - 1)) = sieveA + 1 := by
have hc := upper_integral_one.1
have ha : IntegrableOn (fun s ↦ lowerDeficit (s - 1)) (Ioi (1 : ℝ)) := by
simpa only [zero_add] using integrableOn_shift_Ioi lower_integral_zero.1 1
refine ⟨Iff.mpr integrableOn_Ici_iff_integrableOn_Ioi (hc.add ha), ?_⟩
rw [integral_Ici_eq_integral_Ioi, integral_add hc ha, upper_integral_one.2]
have h := integral_shift_Ioi lowerDeficit 0 1
rw [zero_add] at h
rw [h, lower_integral_zero.2]
ring
end Erdos970Dependency.DeficitIntegralValues
end
end Erdos970
end OAI
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