File size: 5,496 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 82 83 84 85 86 87 88 89 90 | import OAI.Combinatorics.EditApproximation.Binary.BinaryGroupActionSourceSpace
import OAI.Combinatorics.EditApproximation.Binary.BinaryCellSourcePort
import OAI.Combinatorics.EditApproximation.Binary.BinaryUnskippedSpace
import OAI.Combinatorics.EditApproximation.Queries.QueryLocalRepresentativeSpace
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)
(parentAllocation : TargetInterval target.length → ℕ)
(childAllocation initialAllocation currentAllocation : Fin M → TargetInterval target.length → ℕ)
(drawAllocation : ∀ t : Fin (groupDyadicScales M b (nat F).value).length,
Fin (groupSampleCount M Q b (groupDyadicScaleAt M b (nat F).value t)) → ℕ)
def preparedCellGroupSourceAllocation (cell : ℕ × ℕ) : ℕ :=
let centers := (cellCentersReadWithWork P F b a (inversePowerTwo exponent) parentInitial cell).1
let members := (filterMapWithWork
(preparedUnskippedReadWithWork source target parent b P exponent hb hP tau (nat F)
child initial R C hr hi) centers).1
cellCentersReadAllocation P F b a (inversePowerTwo exponent) parentInitial parentAllocation cell +
filterMapAllocation (preparedUnskippedReadWithWork source target parent b P exponent hb hP tau (nat F)
child initial R C hr hi)
(fun center => preparedUnskippedReadAllocation parent center b P exponent tau (nat F)
child initial childAllocation initialAllocation) centers +
if hempty : members = [] then 0 else
sampledGroupActionSourceAllocation source target parent b P exponent Q tau (nat F)
(fun i q => (child i q).1) initial current initialAllocation currentAllocation
(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))
(fun t j => drawAllocation t (Fin.cast (congrArg (groupSampleCount M Q b)
(indexedGroupScales_value raw M b (nat F).value hraw t)) j)) members hempty + 1
theorem preparedCellGroupSourceAllocation_le_work (cell : ℕ × ℕ)
(hp : ∀ q, parentAllocation q ≤ (parentInitial q).2)
(hc : ∀ i q, childAllocation i q ≤ (child i q).2)
(hin : ∀ i q, initialAllocation i q ≤ (initial i q).2)
(hcur : ∀ i q, currentAllocation i q ≤ (current i q).2)
(hd : ∀ t i, drawAllocation t i ≤ (draw t i).2) :
preparedCellGroupSourceAllocation source target parent b P F exponent Q hb hP tau a
parentInitial child initial current R C hr hi raw hraw draw
parentAllocation childAllocation initialAllocation currentAllocation drawAllocation cell ≤
(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 := by
let centers := (cellCentersReadWithWork P F b a (inversePowerTwo exponent) parentInitial cell).1
let members := (filterMapWithWork
(preparedUnskippedReadWithWork source target parent b P exponent hb hP tau (nat F)
child initial R C hr hi) centers).1
have hcenters := cellCentersReadAllocation_le_work P F b a (inversePowerTwo exponent)
parentInitial parentAllocation cell hp
have hmembers := filterMapAllocation_le_work
(preparedUnskippedReadWithWork source target parent b P exponent hb hP tau (nat F)
child initial R C hr hi)
(fun center => preparedUnskippedReadAllocation parent center b P exponent tau (nat F)
child initial childAllocation initialAllocation) centers
(fun center _ => preparedUnskippedReadAllocation_le_work source target parent center b P exponent
hb hP tau (nat F) child initial childAllocation initialAllocation R C hr hi hc hin)
dsimp only [centers] at hmembers
unfold preparedCellGroupSourceAllocation preparedCellGroupSourceWithWork
dsimp only
split_ifs with hempty
· exact Nat.add_le_add (Nat.add_le_add hcenters hmembers) (by decide : 0 ≤ 1)
· have hsample := sampledGroupActionSourceAllocation_le_work source target parent b P exponent Q
tau (nat F) (fun i q => (child i q).1) initial current initialAllocation currentAllocation
(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))
(fun t j => drawAllocation t (Fin.cast (congrArg (groupSampleCount M Q b)
(indexedGroupScales_value raw M b (nat F).value hraw t)) j)) members hempty hin hcur
(fun t j => hd t _)
dsimp only [centers, members] at hmembers hsample
exact (Nat.add_assoc _ _ _).symm.le.trans
(Nat.add_le_add (Nat.add_le_add (Nat.add_le_add hcenters hmembers) hsample)
(by decide : 1 ≤ 2))
end Cell
end EditApproximation.BinaryFraction
end OAI
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