File size: 4,153 Bytes
e63eb7f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 | 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
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