| import OAI.Combinatorics.ExpectationThreshold.RoundingMasses |
| import OAI.Combinatorics.ExpectationThreshold.RoundingLowerBound |
| import OAI.Combinatorics.ExpectationThreshold.RoundingProbabilityContradiction |
| import OAI.Combinatorics.ExpectationThreshold.ColorMean |
| import OAI.Combinatorics.ExpectationThreshold.SelectorMain |
|
|
| namespace OAI |
|
|
| /-! |
| # Actual rounding of an unrestricted fractional cover |
|
|
| The concrete member masses feed the proved multiscale selector. The same |
| independent color law supplies the lower and upper bounds on the subset-product |
| score. No selector hypothesis or bound on support sizes appears in the result. |
| -/ |
|
|
| open scoped BigOperators |
|
|
| noncomputable section |
|
|
| namespace TalagrandThreshold |
|
|
| variable {α : Type*} [Fintype α] [DecidableEq α] |
|
|
| omit [DecidableEq α] in |
| theorem fractionalCover_empty_not_mem {F : Family α} {g : Finset α → ℝ} {r : ℝ} |
| (hr : 0 ≤ r) (hcover : FractionalCover F g) (hcost : fracCost r g ≤ 1 / 2) : |
| (∅ : Finset α) ∉ F := by |
| intro hempty |
| have hcoverEmpty : (1 : ℝ) ≤ g ∅ := by |
| simpa only [Finset.powerset_empty, Finset.sum_singleton] using hcover.2 ∅ hempty |
| have hweight : g ∅ ≤ (1 / 2 : ℝ) := |
| (empty_weight_le_fracCost hr hcover.nonneg).trans hcost |
| exact (by norm_num : ¬ ((1 : ℝ) ≤ 1 / 2)) (hcoverEmpty.trans hweight) |
|
|
| / |
| def fractionalMassFamily (F : Family α) (g : Finset α → ℝ) {r : ℝ} |
| (hr : 0 ≤ r) (hcover : FractionalCover F g) (hcost : fracCost r g ≤ 1 / 2) : |
| Selector.MassFamily F where |
| weight H := roundingMass g H.val |
| nonneg H x := roundingMass_nonneg g H.val hcover.nonneg |
| (memberMass_pos_of_fractionalCover hcover H.property hr hcost) x |
| total H := roundingMass_sum g H.val |
| (memberMass_pos_of_fractionalCover hcover H.property hr hcost) |
| supported H _ hx := roundingMass_eq_zero_of_not_mem g H.val hx |
|
|
| / |
| parameter `r / (25 * 512^4)`. Its support may contain subsets of any size. -/ |
| theorem fractionalCover_rounding (F : Family α) (g : Finset α → ℝ) {r : ℝ} |
| (hr0 : 0 < r) (hr1 : r ≤ 1) (hcover : FractionalCover F g) |
| (hcost : fracCost r g ≤ 1 / 2) : |
| Small (r / ((25 : ℝ) * (512 : ℝ) ^ 4)) F := by |
| classical |
| by_cases hF : F.Nonempty |
| · let p : ℝ := r / ((25 : ℝ) * (512 : ℝ) ^ 4) |
| have hp0 : 0 < p := div_pos hr0 rounding_constant_pos |
| have hp1 : p ≤ 1 := (div_le_one rounding_constant_pos).mpr |
| (hr1.trans (by norm_num : (1 : ℝ) ≤ (25 : ℝ) * (512 : ℝ) ^ 4)) |
| have hnoempty := fractionalCover_empty_not_mem hr0.le hcover hcost |
| let mu := fractionalMassFamily F g hr0.le hcover hcost |
| obtain ⟨s, hs, hnext⟩ := exists_scale hp0 |
| let μ := coloringLaw (α := α) p s hp0 hs |
| let Y : Color s → ℝ := fun c => (512 : ℝ) ^ 4 / (512 : ℝ) ^ (c.val + 1) |
| let Z : Coloring α s → ℝ := fun a => productScore g (fun x => Y (a x)) |
| let P : Coloring α s → Prop := fun a => ∃ H : ↥F, Selector.meanColor mu H a ≤ 2 |
| change Small p F |
| by_contra hnotsmall |
| have hprob : (9 / 10 : ℝ) ≤ μ.prob P := |
| Selector.average_color F hF (fun H hH => members_nonempty hnoempty hH) |
| mu p hp0 hp1 hnotsmall s hs |
| have hY : ∀ c, 0 ≤ Y c := by |
| intro c |
| exact div_nonneg (pow_nonneg (by norm_num) _) (pow_nonneg (by norm_num) _) |
| have hZ : ∀ a, 0 ≤ Z a := fun a => |
| productScore_nonneg hcover.nonneg (fun x => hY (a x)) |
| have hlower : ∀ a, P a → (1 / 4 : ℝ) ≤ Z a := by |
| intro a ha |
| obtain ⟨H, hH⟩ := ha |
| apply productScore_lower_of_mean g H.val (fun x => (a x).val + 1) hcover.nonneg |
| (memberMass_ge_half hcover H.property hr0.le hcost) |
| simpa only [Selector.meanColor, mu, fractionalMassFamily, Nat.cast_add, |
| Nat.cast_one] using hH |
| have hmean : (colorLaw p s hp0 hs).expectation Y ≤ r / 5 := by |
| change (∑ c : Color s, |
| colorProb p s c * ((512 : ℝ) ^ 4 / (512 : ℝ) ^ (c.val + 1))) ≤ r / 5 |
| exact color_mean_le_r_div_five hr0 hs hnext |
| have hupper : μ.expectation Z ≤ (1 / 10 : ℝ) := |
| expectation_productScore_le_tenth (colorLaw p s hp0 hs) g Y |
| hcover.nonneg hY hr0.le hmean hcost |
| exact rounding_expectation_contradiction μ P Z hZ hprob hlower hupper |
| · have hFempty : F = ∅ := Finset.not_nonempty_iff_eq_empty.mp hF |
| rw [hFempty] |
| exact small_empty _ |
|
|
| / |
| theorem fractionalSmall_rounding (F : Family α) {r : ℝ} |
| (hr0 : 0 ≤ r) (hr1 : r ≤ 1) (hF : FractionalSmall r F) : |
| Small (r / ((25 : ℝ) * (512 : ℝ) ^ 4)) F := by |
| obtain ⟨g, hcover, hcost⟩ := hF |
| by_cases hr : r = 0 |
| · subst r |
| rw [zero_div] |
| exact small_zero F (fractionalCover_empty_not_mem le_rfl hcover hcost) |
| · exact fractionalCover_rounding F g (lt_of_le_of_ne hr0 (Ne.symm hr)) hr1 hcover hcost |
|
|
| end TalagrandThreshold |
|
|
| end |
|
|
| end OAI |
|
|
