openAI-math-dataset / lean /OAI /Combinatorics /ExpectationThreshold /SelectorGlobalCostProfiles.lean
| import OAI.Combinatorics.ExpectationThreshold.SelectorDefinitions | |
| import OAI.Combinatorics.ExpectationThreshold.SelectorCountingWordsProducts | |
| import OAI.Combinatorics.ExpectationThreshold.TupleGeometric | |
| namespace OAI | |
| /-! | |
| # Summing the full-profile weights | |
| The reconstruction sets may depend on the entire profile. Once their counting | |
| bound has been replaced by the uniform entropy bound, the remaining full | |
| profiles can be grouped by row counts. The literal word-count weights have | |
| total mass at most one within each such group. | |
| -/ | |
| open scoped BigOperators | |
| namespace TalagrandThreshold.Selector | |
| /-- The full-profile sum satisfies the selector budget for any finite family | |
| whose row-count vectors are nonzero. No independence of the rows of the | |
| profile family is required. -/ | |
| theorem sum_profile_weights_lt_one_twentieth (p : ℝ) (s : ℕ) (hp : 0 < p) | |
| (hs : p * scaleSum s ≤ 1 / 2) (N : Finset (Profile s)) | |
| (hN : ∀ n ∈ N, rowCount n ≠ 0) : | |
| (∑ n ∈ N, | |
| (∏ i : Fin s, ((3 : ℝ) / 64 ^ (i.val + 1)) ^ (rowCount n i)) * | |
| (∏ i : Fin s, ((wordCount (rowCount n i) (n i) : ℝ) * | |
| ∏ h : Color s, colorProb p s h ^ (n i h)))) < 1 / 20 := by | |
| classical | |
| let c : (Fin s → ℕ) → ℝ := fun t => | |
| ∏ i : Fin s, ((3 : ℝ) / 64 ^ (i.val + 1)) ^ (t i) | |
| let w : (Fin s → ℕ) → Profile s → ℝ := fun t n => | |
| ∏ i : Fin s, ((wordCount (t i) (n i) : ℝ) * | |
| ∏ h : Color s, colorProb p s h ^ (n i h)) | |
| let T : Finset (Fin s → ℕ) := N.image rowCount | |
| have hc (t : Fin s → ℕ) : 0 ≤ c t := by | |
| exact Finset.prod_nonneg fun i _ => | |
| pow_nonneg (div_nonneg (by norm_num) (pow_nonneg (by norm_num) _)) _ | |
| have hfiber (t : Fin s → ℕ) : | |
| (∑ n ∈ N.filter (fun n => rowCount n = t), | |
| c (rowCount n) * w (rowCount n) n) ≤ c t := by | |
| have hw : (∑ n ∈ N.filter (fun n => rowCount n = t), w t n) ≤ 1 := | |
| sum_prod_wordCount_weight_le_one t (N.filter (fun n => rowCount n = t)) | |
| (colorProb p s) (colorProb_nonneg p s hp hs) (sum_colorProb p s).le | |
| calc | |
| (∑ n ∈ N.filter (fun n => rowCount n = t), | |
| c (rowCount n) * w (rowCount n) n) = | |
| c t * ∑ n ∈ N.filter (fun n => rowCount n = t), w t n := by | |
| rw [Finset.mul_sum] | |
| apply Finset.sum_congr rfl | |
| intro n hn | |
| rw [(Finset.mem_filter.mp hn).2] | |
| _ ≤ c t * 1 := mul_le_mul_of_nonneg_left hw (hc t) | |
| _ = c t := mul_one _ | |
| have hT : ∀ t ∈ T, t ≠ 0 := by | |
| intro t ht | |
| obtain ⟨n, hn, rfl⟩ := Finset.mem_image.mp ht | |
| exact hN n hn | |
| change (∑ n ∈ N, c (rowCount n) * w (rowCount n) n) < 1 / 20 | |
| calc | |
| (∑ n ∈ N, c (rowCount n) * w (rowCount n) n) = | |
| ∑ t ∈ T, ∑ n ∈ N.filter (fun n => rowCount n = t), | |
| c (rowCount n) * w (rowCount n) n := | |
| (Finset.sum_fiberwise_of_maps_to | |
| (fun n hn => Finset.mem_image_of_mem rowCount hn) | |
| (fun n => c (rowCount n) * w (rowCount n) n)).symm | |
| _ ≤ ∑ t ∈ T, c t := Finset.sum_le_sum fun t _ => hfiber t | |
| _ < 1 / 20 := tuple_geometric_sum_lt_one_twentieth T hT | |
| end TalagrandThreshold.Selector | |
| end OAI | |