openAI-math-dataset / lean /OAI /Combinatorics /BalancedRyser /Profiles /ProfileDetectionSource.lean
| 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 | |