| import OAI.Combinatorics.EditApproximation.Binary.BinaryGroupReadCosts |
| import OAI.Combinatorics.EditApproximation.Binary.BinaryCellReads |
|
|
| namespace OAI |
|
|
| /-! The complete whole-cell estimator, including its ordered choice, |
| executes with a linear factor for each finite-buffer lookup. -/ |
|
|
| namespace EditApproximation |
| namespace BinaryFraction |
|
|
| theorem cellCentersReadWithWork_factor {n : ℕ} (P F b : ℕ) (a theta : BinaryFraction) |
| (initial : TargetInterval n → BinaryFraction × ℕ) (cell : ℕ × ℕ) |
| (R : ℕ) (hr : ∀ r, (initial r).2 ≤ R) : |
| (cellCentersReadWithWork P F b a theta initial cell).2 ≤ |
| (cellCentersWithWork P F b a theta (fun r => (initial r).1) cell).2 * tableReadFactor R := by |
| have h := (cellCentersReadWithWork_eq_cost P F b a theta initial cell R hr).2 |
| have hK : R + 1 ≤ tableReadFactor R := by unfold tableReadFactor; omega |
| exact h.trans (Nat.mul_le_mul_left _ hK) |
|
|
| section Cell |
| variable {α : Type*} (source target : List α) {M : ℕ} |
| (parent : TargetInterval source.length) (b P F exponent Q : ℕ) (hb : 0 < b) (hP : 0 < P) |
| (tau a : BinaryFraction) |
| (parentInitial : TargetInterval target.length → BinaryFraction × ℕ) |
| (child initial current : Fin M → TargetInterval target.length → BinaryFraction × ℕ) |
| (R : ℕ) (hc : ∀ i q, (child i q).2 ≤ R) (hi : ∀ i q, (initial i q).2 ≤ R) |
| (raw : List BinaryFraction) (hraw : raw.map value = groupDyadicScales M b (nat F).value) |
| (draw : ∀ t : Fin (groupDyadicScales M b (nat F).value).length, |
| Fin (groupSampleCount M Q b (groupDyadicScaleAt M b (nat F).value t)) → Fin M) |
|
|
| theorem preparedCellGroupReadWithWork_factor (cell : ℕ × ℕ) |
| (hp : ∀ r, (parentInitial r).2 ≤ R) (hn : ∀ i q, (current i q).2 ≤ R) : |
| (preparedCellGroupReadWithWork source target parent b P F exponent Q hb hP tau a |
| parentInitial child initial current R R hc hi raw hraw draw cell).2 ≤ |
| (preparedCellGroupWithWork source target parent b P F exponent Q hb tau a |
| (fun r => (parentInitial r).1) (fun i q => (child i q).1) |
| (fun i q => (initial i q).1) (fun i q => (current i q).1) raw hraw draw hP cell).2 * |
| tableReadFactor R := by |
| have hcenters := cellCentersReadWithWork_factor P F b a (inversePowerTwo exponent) |
| parentInitial cell R hp |
| have hcentersq := (cellCentersReadWithWork_eq_cost P F b a (inversePowerTwo exponent) |
| parentInitial cell R hp).1 |
| have hm := filterMapWithWork_factor |
| (preparedUnskippedReadWithWork source target parent b P exponent hb hP tau (nat F) |
| child initial R R hc hi) |
| (preparedUnskippedCandidateWithWork source target parent b P exponent hb hP tau (nat F) |
| (fun i q => (child i q).1) (fun i q => (initial i q).1)) |
| (cellCentersWithWork P F b a (inversePowerTwo exponent) (fun r => (parentInitial r).1) cell).1 |
| (tableReadFactor R) (tableReadFactor_pos R) |
| (fun center _ => preparedUnskippedReadWithWork_value source target parent b P exponent hb hP |
| tau (nat F) child initial R R hc hi center) |
| (fun center _ => preparedUnskippedReadWithWork_factor source target parent b P exponent hb hP |
| tau (nat F) child initial R hc hi center) |
| have hcm := Nat.add_le_add hcenters hm.2 |
| rw [← Nat.add_mul] at hcm |
| unfold preparedCellGroupReadWithWork preparedCellGroupWithWork |
| dsimp only |
| simp only [hcentersq, hm.1] |
| split_ifs with hempty |
| · apply cost_replace_factor _ _ _ _ _ hcm (tableReadFactor_pos R) <;> omega |
| · have hs := sampledGroupActionReadWithWork_factor source target parent b P exponent Q tau |
| (nat F) (fun i q => (child i q).1) initial current |
| (indexedGroupScales raw M b (nat F).value hraw) |
| (fun t j => draw t (Fin.cast (congrArg (groupSampleCount M Q b) |
| (indexedGroupScales_value raw M b (nat F).value hraw t)) j)) |
| _ hempty R hi hn |
| have h := Nat.add_le_add hcm hs |
| rw [← Nat.add_mul] at h |
| change _ ≤ _ at hs |
| unfold dyadicSampledGroupActionReadWithWork dyadicSampledGroupActionWithWork |
| dsimp only |
| apply cost_replace_factor _ _ _ _ _ h (tableReadFactor_pos R) <;> omega |
|
|
| end Cell |
| end BinaryFraction |
| end EditApproximation |
|
|
| end OAI |
|
|
