File size: 4,693 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 81 | import OAI.Combinatorics.EditApproximation.Binary.BinarySampledSourcePort
import OAI.Combinatorics.EditApproximation.Binary.BinaryCellReads
namespace OAI
namespace EditApproximation.BinaryFraction
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 C : ℕ) (hr : ∀ i q, (child i q).2 ≤ R) (hi : ∀ i q, (initial i q).2 ≤ C)
(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 × ℕ)
/-- The same raw dyadic scale list and Fin-count indices as the original
cell estimator, now with the cost of decoding every sampled coordinate. -/
def preparedCellGroupSourceWithWork (cell : ℕ × ℕ) :
Option (BinaryBellmanAction (TargetInterval target.length)
(Fin M × TargetInterval target.length) M) × ℕ :=
let centers := cellCentersReadWithWork P F b a (inversePowerTwo exponent) parentInitial cell
let members := filterMapWithWork
(preparedUnskippedReadWithWork source target parent b P exponent hb hP tau (nat F)
child initial R C hr hi) centers.1
if h : members.1 = [] then (none, centers.2 + members.2 + 1) else
let selected := sampledGroupActionSourceWithWork 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)) members.1 h
(some selected.1, centers.2 + members.2 + selected.2 + 2)
theorem preparedCellGroupSourceWithWork_eq_cost (cell : ℕ × ℕ) (S : ℕ)
(hs : ∀ t i, (draw t i).2 ≤ S) :
(preparedCellGroupSourceWithWork source target parent b P F exponent Q hb hP tau a
parentInitial child initial current R C hr hi raw hraw draw cell).1 =
(preparedCellGroupReadWithWork source target parent b P F exponent Q hb hP tau a
parentInitial child initial current R C hr hi raw hraw (fun t i => (draw t i).1) cell).1 ∧
(preparedCellGroupSourceWithWork source target parent b P F exponent Q hb hP tau a
parentInitial child initial current R C hr hi raw hraw draw cell).2 ≤
(preparedCellGroupReadWithWork source target parent b P F exponent Q hb hP tau a
parentInitial child initial current R C hr hi raw hraw (fun t i => (draw t i).1) cell).2 * (S + 4) := by
let centers := cellCentersReadWithWork P F b a (inversePowerTwo exponent) parentInitial cell
let members := filterMapWithWork
(preparedUnskippedReadWithWork source target parent b P exponent hb hP tau (nat F)
child initial R C hr hi) centers.1
unfold preparedCellGroupSourceWithWork preparedCellGroupReadWithWork
dsimp only
split_ifs with hempty
· exact ⟨rfl, by dsimp only; nlinarith only [Nat.zero_le S]⟩
· have h := sampledGroupActionSourceWithWork_eq_cost 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)) members.1 hempty S
(fun t j => hs t _)
dsimp only [dyadicSampledGroupActionReadWithWork]
rw [h.1]
exact ⟨rfl, by nlinarith only [h.2]⟩
theorem preparedCellGroupSourceWithWork_value (cell : ℕ × ℕ) :
(preparedCellGroupSourceWithWork source target parent b P F exponent Q hb hP tau a
parentInitial child initial current R C hr hi raw hraw draw cell).1 =
(preparedCellGroupReadWithWork source target parent b P F exponent Q hb hP tau a
parentInitial child initial current R C hr hi raw hraw (fun t i => (draw t i).1) cell).1 := by
unfold preparedCellGroupSourceWithWork preparedCellGroupReadWithWork
dsimp only
split_ifs with hempty
· rfl
· apply congrArg some
exact sampledGroupActionSourceWithWork_value 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
end Cell
end EditApproximation.BinaryFraction
end OAI
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