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159357f | 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 95 96 97 98 | import Mathlib
import OAI.Analysis.CoulombIonization.RadialBounds.ActualBandInverseBarrier
import OAI.Analysis.CoulombIonization.RadialBounds.OwnProbabilityRareCapBarrier
namespace OAI
noncomputable section
open MeasureTheory Filter
open scoped Topology BigOperators ContDiff
open MeasureTheory Filter Set Metric
open scoped Topology BigOperators
namespace CoulombAtom
open CoulombAnalysis CoulombObservation CoulombBarrier
attribute [local irreducible] graphComponent graphFormVector fermionGraph weakGraph fermionGraphValue
attribute [local irreducible] physicalObservationLaw jointMasterPosterior
lemma originalQueryDensity_info_measurable {N K : ℕ} (F : fermionGraph N)
(r : ℝ) (j : ℕ) {c₁ r₀ s : ℝ} (hc : 0 < c₁) (hr₀ : 0 < r₀)
(hs : 0 < s) (y : Space) :
Measurable[observationInformation (fun k : Fin K => dyadicObservationWidth r k) j]
(fun z => originalQueryDensity F r j c₁ r₀ s z y) := by
exact (jointMasterPosterior_measurable (graphRawLaw F)
(fun k : Fin K => dyadicObservationWidth r k) j hc hr₀ hs
canonicalRealPacket_smooth.continuous).comp (measurable_id.prodMk measurable_const)
lemma original_inverse_failure_info_measurable {N K : ℕ} (F : fermionGraph N)
(Z lam r₀ s c₁ h xi : ℝ) (j : Fin K) (y : Space)
(hc : 0 < c₁) (hr₀ : 0 < r₀) (hs : 0 < s) :
MeasurableSet[observationInformation (fun k : Fin K => dyadicObservationWidth r₀ k) j]
(originalLowFailure F Z lam r₀ s c₁ h xi j y ∪
originalHighFailure F Z lam r₀ s c₁ h xi j y) := by
have hf := originalQueryField_info_measurable (K := K) F Z lam r₀ j hc hr₀ hs y
have hd := originalQueryDensity_info_measurable (K := K) F r₀ j hc hr₀ hs y
exact ((measurableSet_le measurable_const hf).inter (measurableSet_le hd measurable_const)).union
((measurableSet_le hf measurable_const).inter (measurableSet_le measurable_const hd))
lemma original_inverse_height_net_union_bound {N K : ℕ} {F : fermionGraph N}
{Z lam r₀ s c₁ xi : ℝ} {j : Fin K} (Q : Finset Space) (H : Finset ℝ)
(hQ : ∀ h ∈ H, ∀ y ∈ Q, OriginalPointInverseBounds F Z lam r₀ s c₁ h xi j y) :
(physicalObservationLaw (graphRawLaw F) K).real
(⋃ h ∈ H, ⋃ y ∈ Q, originalLowFailure F Z lam r₀ s c₁ h xi j y ∪
originalHighFailure F Z lam r₀ s c₁ h xi j y) ≤
4*(H.card:ℝ)*(Q.card:ℝ)*((2:ℝ)^j.val*r₀)^40 := by
apply (measureReal_biUnion_finset_le H _).trans
calc
_ ≤ ∑ _h ∈ H, 4*(Q.card:ℝ)*((2:ℝ)^j.val*r₀)^40 := by
exact Finset.sum_le_sum (fun h hh => original_inverse_net_union_bound Q (hQ h hh))
_ = _ := by simp only [Finset.sum_const, nsmul_eq_mul]; ring
lemma exists_inverse_band_spatial_net {u : ℝ} (hu : 0 < u) (hu1 : u ≤ 1) :
∃ Q : Finset Space,
(∀ x ∈ Q, 11*u/8 ≤ ‖x‖ ∧ ‖x‖ ≤ 13*u/8) ∧
(∀ y : Space, 11*u/8 ≤ ‖y‖ → ‖y‖ ≤ 13*u/8 →
∃ x ∈ Q, ‖x-y‖ ≤ u^2) ∧
(Q.card:ℝ) ≤ 4913/u^3 := by
obtain ⟨Q,hQ,hcover,hcard⟩ := exists_spatial_grid_net
(A := {x : Space | 11*u/8 ≤ ‖x‖ ∧ ‖x‖ ≤ 13*u/8})
(R := 2*u) (h := u^2/3) (by positivity) (by positivity)
(fun _ hx => by linarith [hx.2])
refine ⟨Q,hQ,?_,hcard.trans ?_⟩
· intro y hyl hyh
obtain ⟨x,hx,hxy⟩ := hcover y ⟨hyl,hyh⟩
refine ⟨x,hx,?_⟩
rw [norm_sub_rev] at hxy
nlinarith
· have hlin : 2*(2*u/(u^2/3))+5 ≤ 17/u := by
apply (le_div_iff₀ hu).mpr
field_simp
nlinarith
calc _ ≤ (17/u)^3 := pow_le_pow_left₀ (by positivity) hlin 3
_ = _ := by ring
lemma divided_net_field_variation {a b alpha beta t : ℝ}
(ht : 1 ≤ t) (ha : 0 ≤ alpha) (hv : |b-a| ≤ alpha+beta*max (-a) 0) :
|b/t-a/t| ≤ alpha+beta*max (-(a/t)) 0 := by
have ht0 : 0 < t := lt_of_lt_of_le zero_lt_one ht
have hm : max (-(a/t)) 0 = max (-a) 0/t := by
simpa only [neg_div,zero_div] using max_div_div_right ht0.le (-a) 0
rw [←sub_div,abs_div,abs_of_pos ht0,hm]
calc
_ ≤ (alpha+beta*max (-a) 0)/t := div_le_div_of_nonneg_right hv ht0.le
_ = alpha/t+beta*(max (-a) 0/t) := by ring
_ ≤ _ := add_le_add (div_le_self ha ht) le_rfl
lemma localCellRadius_normalized {y : Space} (p : ℕ) (a : ℝ) :
(localCellRadius y)^p*a = (‖y‖^p*a)/(100000:ℝ)^p := by
simp only [localCellRadius,div_pow]
ring
end CoulombAtom
end
end OAI
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