openAI-math-dataset / lean /OAI /Combinatorics /EditApproximation /Coarse /CoarseFirstRequestCost.lean
| import OAI.Combinatorics.EditApproximation.Refinement.ComputedCoarseWidthCost | |
| import OAI.Combinatorics.EditApproximation.Probability.FreshCoordinateCost | |
| import OAI.Combinatorics.EditApproximation.Coarse.CoarseGlobalSeed | |
| namespace OAI | |
| /-! First-request work for a complete independent coarse source. The | |
| query is adaptive; each entry is charged only on its first appearance. | |
| No conditioning on seed correctness occurs in this estimate. -/ | |
| namespace EditApproximation | |
| open MeasureTheory | |
| section FirstRequest | |
| variable {ι α : Type*} [Fintype ι] [DecidableEq ι] [DecidableEq α] | |
| (N B : ℕ) [NeZero B] [MeasurableSpace (Fin B)] [MeasurableSingletonClass (Fin B)] | |
| (depth : ι → ℕ) (x y : ι → List α) (query : FiniteQuery ι) | |
| noncomputable def coarseFirstRequestWork (draw : CoarseFullSeedDraws B depth x y) : ℝ := | |
| ∑ i, selectionIndicator (i ∈ query.keys | |
| (fun j => (coarseFullSeedTable N B depth x y draw j : ℚ))) * | |
| (coarseInitialArrayWork B (depth i) (initialThresholdQ N) (initialLambda N) (x i) (y i) (draw i) : ℝ) | |
| theorem coarseFirstRequestWork_expected_le | |
| (hB : B = (integerParameters N).B) (hN : 0 < N) (hell : 200 ≤ inputSmallLog N) | |
| (hdepth : ∀ i, depth i = computedCoarseDepth N (coarseEntryWidth (x i) (y i))) | |
| (hsize : ∀ i, coarseEntryWidth (x i) (y i) ≤ N) | |
| (hinverse : coarseInverseAllowanceFactor B (initialThresholdQ N) (initialLambda N) ≤ | |
| (inputHeight N : ℝ) ^ 8) | |
| (budget : ℝ) | |
| (hbudget : ∀ draw : CoarseFullSeedDraws B depth x y, | |
| ((query.cost (fun i => (coarseFullSeedTable N B depth x y draw i : ℚ)) | |
| (fun i => (x i).length) : ℕ) : ℝ) ≤ budget) : | |
| (∫ draw, coarseFirstRequestWork N B depth x y query draw | |
| ∂coarseFullSeedMeasure B depth x y) ≤ budget * computedCoarseWidthOverhead N := by | |
| let μ := fun i => coarseInitialMeasure B (depth i) (coarseEntryWidth (x i) (y i)) | |
| let answer := fun i draw => | |
| (coarseInitialValue B (depth i) (initialThresholdQ N) (initialLambda N) N (x i) (y i) draw : ℚ) | |
| let cost := fun i draw => | |
| (coarseInitialArrayWork B (depth i) (initialThresholdQ N) (initialLambda N) (x i) (y i) draw : ℝ) | |
| have hcost (i) : (∫ value, cost i value ∂μ i) ≤ | |
| ((x i).length : ℝ) * computedCoarseWidthOverhead N := | |
| computed_coarse_expected_source_width N B hB hN hell hinverse (x i) (y i) | |
| (depth i) (hdepth i) (hsize i) | |
| have hselected (draw : CoarseFullSeedDraws B depth x y) : | |
| (∑ i, selectionIndicator (i ∈ query.keys (fun j => answer j (draw j))) * | |
| ((x i).length : ℝ)) ≤ budget := by | |
| have hs := sum_selected_nat_weights_le (query.keys (fun j => answer j (draw j))) | |
| (fun i => (x i).length) | |
| have hb := hbudget draw | |
| rw [FiniteQuery.cost_eq_sum] at hb | |
| exact hs.trans hb | |
| have h := finiteQuery_fresh_expected_cost_le μ query answer cost | |
| (fun i => ((x i).length : ℝ)) (computedCoarseWidthOverhead N) budget | |
| (Nat.cast_nonneg _) hcost hselected | |
| exact h | |
| end FirstRequest | |
| end EditApproximation | |
| end OAI | |