| import Mathlib.MeasureTheory.Group.Arithmetic |
| import Mathlib.MeasureTheory.Integral.DominatedConvergence |
| import OAI.Combinatorics.BalancedRyser.Profiles.ProfileMeans |
| import OAI.Combinatorics.BalancedRyser.Selection.OccupancyVisibility |
|
|
| namespace OAI |
|
|
| namespace Balanced.OccupancyVisibility |
|
|
| open MeasureTheory |
|
|
| theorem row_summable_of_probability (p : Profile) |
| (hp : HasSum p 1) (i : ℕ) : Summable (fun j : ℕ => p (i, j)) := |
| hp.summable.comp_injective (fun _ _ h => Prod.mk.inj h |>.2) |
|
|
| theorem row_mass_le_one (p : Profile) (hp : HasSum p 1) |
| (hnonneg : ∀ ij, 0 ≤ p ij) (i : ℕ) : (∑' j : ℕ, p (i, j)) ≤ 1 := by |
| simpa only [Function.comp_def, hp.tsum_eq] using |
| tsum_comp_le_tsum_of_inj hp.summable hnonneg |
| (i := fun j : ℕ => (i, j)) (fun _ _ h => Prod.mk.inj h |>.2) |
|
|
| theorem integrable_row_of_probability_support |
| (ν : Measure Profile) [IsFiniteMeasure ν] (i : ℕ) |
| (hsupport : ∀ᵐ x : Profile ∂ν, (∀ ij, 0 ≤ x ij) ∧ HasSum x 1) : |
| Integrable (fun x : Profile => ∑' j : ℕ, x (i, j)) ν := by |
| apply (integrable_const (1 : ℝ)).mono' |
| (Measurable.tsum (fun j => measurable_pi_apply (i, j))).aestronglyMeasurable |
| filter_upwards [hsupport] with x hx |
| rw [Real.norm_eq_abs, abs_of_nonneg (tsum_nonneg (fun j => hx.1 (i, j)))] |
| exact row_mass_le_one x hx.2 hx.1 i |
|
|
| theorem row_integral_hasSum |
| (ν : Measure Profile) [IsFiniteMeasure ν] (i : ℕ) |
| (hsupport : ∀ᵐ x : Profile ∂ν, (∀ ij, 0 ≤ x ij) ∧ HasSum x 1) : |
| HasSum (fun j : ℕ => ∫ x : Profile, x (i, j) ∂ν) |
| (∫ x : Profile, ∑' j : ℕ, x (i, j) ∂ν) := by |
| apply hasSum_integral_of_dominated_convergence (fun j (x : Profile) => x (i, j)) |
| · intro j |
| exact (measurable_pi_apply (i, j)).aestronglyMeasurable |
| · intro j |
| filter_upwards [hsupport] with x hx |
| rw [Real.norm_eq_abs, abs_of_nonneg (hx.1 (i, j))] |
| · filter_upwards [hsupport] with x hx |
| exact row_summable_of_probability x hx.2 i |
| · exact integrable_row_of_probability_support ν i hsupport |
| · filter_upwards [hsupport] with x hx |
| exact (row_summable_of_probability x hx.2 i).hasSum |
|
|
| theorem row_integral_eq_weighted_row |
| (P : Profile) (ν : Measure Profile) [IsFiniteMeasure ν] |
| (hsupport : ∀ᵐ x : Profile ∂ν, (∀ ij, 0 ≤ x ij) ∧ HasSum x 1) |
| (hinc : ∀ j : ℕ, (∫ x : Profile, x (0, j) ∂ν) = (j : ℝ) * P (0, j)) : |
| (∫ x : Profile, ∑' j : ℕ, x (0, j) ∂ν) = ∑' j : ℕ, (j : ℝ) * P (0, j) := by |
| have h := row_integral_hasSum ν 0 hsupport |
| simpa only [hinc] using h.tsum_eq.symm |
|
|
| theorem visible_excess_of_normalized_common_sign |
| (P : Profile) (νD νC : Measure Profile) |
| [IsFiniteMeasure νD] [IsFiniteMeasure νC] |
| {k c lam : ℝ} (hc : 0 < c) (hlam : 0 < lam) |
| (hmassD : νD.real Set.univ = k) (hmassC : νC.real Set.univ = c) |
| (hincD : ∀ i j : ℕ, (∫ x, x (i + 1, j) ∂νD) = |
| (i + 1 : ℝ) * P (i + 1, j)) |
| (hincC : ∀ ij : ℕ × ℕ, (∫ x, x ij ∂νC) = (ij.2 : ℝ) * P ij) |
| (hsupport : ∀ᵐ x : Profile ∂νC, (∀ ij, 0 ≤ x ij) ∧ HasSum x 1) |
| (hP : HasSum (fun j : ℕ => P (0, j)) (Real.exp (-k))) |
| (hdev : (∀ᵐ x : Profile ∂νC, Real.exp (-k) + lam ≤ ∑' j : ℕ, x (0, j)) ∨ |
| (∀ᵐ x : Profile ∂νC, (∑' j : ℕ, x (0, j)) ≤ Real.exp (-k) - lam)) : |
| c < νD.real (deletionVisible P) + νC.real (coverVisible P) := by |
| have hmean : HasSum (fun ij : ℕ × ℕ => (ij.2 : ℝ) * P ij) c := by |
| apply deletion_profile_second_mean νC (fun ij (x : Profile) => x ij) P c |
| · intro ij |
| exact (measurable_pi_apply ij).aestronglyMeasurable |
| · intro ij |
| exact hsupport.mono (fun x hx => hx.1 ij) |
| · exact hsupport.mono (fun x hx => hx.2) |
| · exact hincC |
| · exact hmassC |
| exact visible_excess_of_common_sign P νD νC hc hlam hmassD hmassC hincD |
| hmean hP (integrable_row_of_probability_support νC 0 hsupport) |
| (row_integral_eq_weighted_row P νC hsupport (fun j => hincC (0, j))) hdev |
|
|
| end Balanced.OccupancyVisibility |
|
|
| end OAI |
|
|
