| import OAI.Combinatorics.BalancedRyser.Profiles.AnalyticDetection |
| import OAI.Combinatorics.BalancedRyser.Profiles.ProfileIdentifiability |
|
|
| namespace OAI |
|
|
| open MeasureTheory Set |
|
|
| namespace Balanced |
| open scoped _root_.Balanced |
|
|
| lemma power_series_equality_null {ι : Type*} (a b rate : ι → ℝ) |
| (ha : Summable fun i ↦ |a i|) (hb : Summable fun i ↦ |b i|) |
| (hrate : ∀ i, 0 ≤ rate i) (hinj : Function.Injective rate) |
| (hfinite : ∀ t : ℝ, {i | rate i ≤ t}.Finite) (hne : a ≠ b) : |
| volume.restrict (Ioo (0 : ℝ) 1) |
| {z | (∑' i, a i * z ^ rate i) = ∑' i, b i * z ^ rate i} = 0 := by |
| apply analytic_equality_null (analytic_real_power_series a rate ha hrate) |
| (analytic_real_power_series b rate hb hrate) isPreconnected_Ioo measurableSet_Ioo |
| intro heq |
| apply hne |
| apply ProfileIdentifiability.laplaceSeries_injective a b rate ha hb hrate hinj hfinite |
| intro s hs |
| have hz : Real.exp (-s) ∈ Ioo (0 : ℝ) 1 := |
| ⟨Real.exp_pos _, Real.exp_lt_one_iff.mpr (neg_neg_of_pos hs)⟩ |
| have h := heq hz |
| change (∑' i, a i * (Real.exp (-s)) ^ rate i) = |
| ∑' i, b i * (Real.exp (-s)) ^ rate i at h |
| simpa only [power_series_at_exp_neg, ProfileIdentifiability.laplaceSeries] using h |
|
|
| end Balanced |
|
|
| end OAI |
|
|
