openAI-math-dataset / lean /OAI /Combinatorics /EditApproximation /Queries /QueryFamilyReaderBounds.lean
| import OAI.Combinatorics.EditApproximation.Queries.QueryFamilySourceCosts | |
| import OAI.Combinatorics.EditApproximation.Queries.QuerySourceKeyWidths | |
| namespace OAI | |
| namespace EditApproximation.BinaryFraction | |
| def queryFamilySourceReadBudget (J B W : ℕ) : ℕ := | |
| 200 * (1000 * (B + 1) + 1000 + 2) ^ 2 + 120 * (B + 1) ^ 2 + 10 * J + 100 + W | |
| end EditApproximation.BinaryFraction | |
| namespace EditApproximation | |
| section Family | |
| variable {α : Type*} {M J R : ℕ} (target : List α) | |
| (N P F exponent H Q passes copies pass copy B W : ℕ) | |
| (oracle : PhysicalFamilyScalarKey (M := M) (J := J) (R := R) | |
| target N P F exponent H Q passes copies → BinaryFraction) | |
| (readWork : PhysicalFamilyScalarKey (M := M) (J := J) (R := R) | |
| target N P F exponent H Q passes copies → ℕ) | |
| (hN : Nat.size N ≤ B) (hp : Nat.size pass ≤ B) (hc : Nat.size copy ≤ B) | |
| (hps : Nat.size (passes + 1) ≤ B) (hcs : Nat.size (copies + 1) ≤ B) | |
| (hW : ∀ key, readWork key ≤ W) | |
| include hN hp hc hps hcs hW in | |
| theorem queryFamilyGroupRead_bounded_work | |
| (hn : Nat.size target.length ≤ B) (he : exponent + 1 ≤ B) | |
| (node : PhysicalInternalNode M J) (time : Fin R) (key : GroupScalarKey M F Q) | |
| (hk : queryGroupKeySize key ≤ 40 * B + 500) : | |
| (queryFamilyGroupRead target N P F exponent H Q passes copies pass copy node time key).work oracle readWork ≤ | |
| BinaryFraction.queryFamilySourceReadBudget J B W := by | |
| let D := 1000 * (B + 1) + 1000 | |
| let K := 120 * (B + 1) ^ 2 + 5 * J + 55 | |
| have hb : B ≤ D := by dsimp only [D]; omega | |
| have hkey : queryGroupKeySize key ≤ D := by dsimp only [D]; omega | |
| have hscale := queryScalePresentWithWork_cost N key.1.1 D (hN.trans hb) | |
| (by unfold queryGroupKeySize at hkey; omega) | |
| have hcell := queryCellValidWithWork_cost target.length exponent D key.1.2 | |
| (hn.trans hb) (he.trans hb) (by unfold queryGroupKeySize at hkey; omega) | |
| (by unfold queryGroupKeySize at hkey; omega) | |
| have hr := queryFamilyGroupRead_work target N P F exponent H Q passes copies pass copy oracle readWork K W | |
| (fun scalar => queryFamilyTagWithWork_cost target N P F exponent H Q passes copies pass copy B hp hc hps hcs scalar) | |
| hW node time key | |
| change _ ≤ 200 * (D + 2) ^ 2 + 120 * (B + 1) ^ 2 + 10 * J + 100 + W | |
| dsimp only [K] at hr | |
| nlinarith only [hscale, hcell, hr, Nat.zero_le D] | |
| include hN hp hc hps hcs hW in | |
| theorem queryFamilyOnlineRead_bounded_work | |
| (hbound : Nat.size (localOnlineMassBound M P H) ≤ B) | |
| (node : PhysicalInternalNode M J) (time : Fin R) (key : ℕ × TargetInterval target.length) | |
| (range : ℕ) (hk : Nat.size key.1 ≤ B + 1) (hrange : Nat.size range ≤ 240 * (B + 5) + 7) : | |
| (queryFamilyOnlineRead target N P F exponent H Q passes copies pass copy node time key range).work oracle readWork ≤ | |
| BinaryFraction.queryFamilySourceReadBudget J B W := by | |
| let D := 1000 * (B + 1) + 1000 | |
| let K := 120 * (B + 1) ^ 2 + 5 * J + 55 | |
| have hb : B ≤ D := by dsimp only [D]; omega | |
| have hs := queryScalePresentWithWork_cost N key.1 D (hN.trans hb) (by dsimp only [D]; omega) | |
| have hv := queryRangeValidWithWork_cost (localOnlineMassBound M P H) range D | |
| (hbound.trans hb) (by dsimp only [D]; omega) | |
| have hr := queryFamilyOnlineRead_work target N P F exponent H Q passes copies pass copy oracle readWork K W | |
| (fun scalar => queryFamilyTagWithWork_cost target N P F exponent H Q passes copies pass copy B hp hc hps hcs scalar) | |
| hW node time key range | |
| change _ ≤ 200 * (D + 2) ^ 2 + 120 * (B + 1) ^ 2 + 10 * J + 100 + W | |
| dsimp only [K] at hr | |
| nlinarith only [hs, hv, hr, Nat.zero_le D] | |
| end Family | |
| end EditApproximation | |
| end OAI | |