File size: 2,451 Bytes
f0f05de | 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 | import OAI.Combinatorics.BalancedRyser.Profiles.ProfileDetectionLimit
import OAI.Combinatorics.BalancedRyser.Profiles.ProfileEmpiricalSupport
namespace OAI
open MeasureTheory Filter
open scoped Topology
namespace Balanced.ProfileDetectionSource
open ProfileDetection ProfileDetectionBridge ProfileParameterLaw ProfileDetectionLimit
universe u
theorem profileSequenceLimit_cutoff_sum_visible_mass
{F : ℕ → Type u} [∀ n, Field (F n)] [∀ n, Fintype (F n)]
[∀ n, DecidableEq (F n)]
(k : ℕ) (D C : ∀ n, Finset (Line (F n)))
(hpair : ∀ n, IsProfilePair k (D n) (C n))
{c : ℝ} {P : Profile} {νD νC : FiniteMeasure Profile}
(ht : ProfileSequenceLimit D C c P νD νC) :
Tendsto (fun n : ℕ =>
((νD : Measure Profile).prod (parameterLaw 1)).real
(cutoffDetection Prod.snd (dDiscrepancy P) n) +
((νC : Measure Profile).prod (parameterLaw (Real.sqrt 2))).real
(cutoffDetection Prod.snd (cDiscrepancy P) n)) atTop
(𝓝 ((νD : Measure Profile).real (deletionVisible P) +
(νC : Measure Profile).real (coverVisible P))) := by
obtain ⟨hP, hD, hC⟩ :=
ProfileEmpiricalSupport.profileSequenceLimit_bounded k D C hpair ht
obtain ⟨hD0, hC0⟩ :=
ProfileEmpiricalSupport.profileSequenceLimit_zero_coordinates k D C hpair ht
apply tendsto_cutoff_sum_visible_mass νD νC P hP.2.1.summable
· filter_upwards [hD, hD0] with x hx hx0
exact ⟨hx.2.1.summable, hx0⟩
· filter_upwards [hC, hC0] with x hx hx0
exact ⟨hx.2.1.summable, hx0⟩
theorem exists_cutoff_of_visible_excess
{F : ℕ → Type u} [∀ n, Field (F n)] [∀ n, Fintype (F n)]
[∀ n, DecidableEq (F n)]
(k : ℕ) (D C : ∀ n, Finset (Line (F n)))
(hpair : ∀ n, IsProfilePair k (D n) (C n))
{c : ℝ} {P : Profile} {νD νC : FiniteMeasure Profile}
(ht : ProfileSequenceLimit D C c P νD νC)
(hexcess : c < (νD : Measure Profile).real (deletionVisible P) +
(νC : Measure Profile).real (coverVisible P)) :
∃ n : ℕ, c <
((νD : Measure Profile).prod (parameterLaw 1)).real
(cutoffDetection Prod.snd (dDiscrepancy P) n) +
((νC : Measure Profile).prod (parameterLaw (Real.sqrt 2))).real
(cutoffDetection Prod.snd (cDiscrepancy P) n) := by
exact ((tendsto_order.mp
(profileSequenceLimit_cutoff_sum_visible_mass k D C hpair ht)).1 c hexcess).exists
end Balanced.ProfileDetectionSource
end OAI
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