File size: 3,944 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 | import OAI.Combinatorics.EditApproximation.Binary.BinarySelectionSpace
import OAI.Combinatorics.EditApproximation.Binary.BinaryBandActionReads
import OAI.Combinatorics.EditApproximation.Queries.QueryBufferSpace
import OAI.Combinatorics.EditApproximation.Binary.BinaryOnlineSpace
import OAI.Combinatorics.EditApproximation.Definitions
namespace OAI
namespace EditApproximation.BinaryFraction
theorem countedBandCoordinateAllocation_le_work {M n : ℕ} (mass : Vector BinaryFraction M)
(total : BinaryFraction) (action : Fin M → TargetInterval n) (coordinate : Fin M × TargetInterval n) :
countedBandCoordinateAllocation mass total action coordinate ≤
(countedBandCoordinateWithWork mass total action coordinate).2 := by
have hc := coordinateEqualAllocation_le_work coordinate (coordinate.1, action coordinate.1)
have hd := divAllocation_le_work (mass.get coordinate.1) total
unfold countedBandCoordinateAllocation countedBandCoordinateWithWork
dsimp only
split_ifs <;> (try dsimp only) <;> omega
theorem countedBandVerticesAllocation_le_work {M n d : ℕ} (coordinates : Vector (Fin M × TargetInterval n) d)
(mass : Vector BinaryFraction M) (band : List (Fin M → TargetInterval n)) :
countedBandVerticesAllocation coordinates mass band ≤ (countedBandVerticesWithWork coordinates mass band).2 := by
have hm := bandMassAllocation_le_work mass
have hs : (band.map fun action => vectorMapAllocation d fun i =>
countedBandCoordinateAllocation mass (bandMassTotalWithWork mass).1 action (coordinates.get i)).sum ≤
(band.map fun action => (vectorMapWithWork d fun i =>
countedBandCoordinateWithWork mass (bandMassTotalWithWork mass).1 action (coordinates.get i)).2).sum := by
induction band with
| nil => simp
| cons a rest ih =>
have h := vectorMapAllocation_le_work d
(fun i => countedBandCoordinateWithWork mass (bandMassTotalWithWork mass).1 a (coordinates.get i))
(fun i => countedBandCoordinateAllocation mass (bandMassTotalWithWork mass).1 a (coordinates.get i))
(fun i => countedBandCoordinateAllocation_le_work _ _ _ _)
simp only [List.map_cons, List.sum_cons]
omega
unfold countedBandVerticesAllocation countedBandVerticesWithWork
simp only [List.map_map, Function.comp_def]
omega
/-- Arithmetic and labeled-vector allocation after coordinate construction.
Coordinate construction itself is accounted by its separate list allocator. -/
theorem countedPreparedActionKernelAllocation_le_work {M n : ℕ} (mass : Vector BinaryFraction M)
(read : ℕ → Fin M × TargetInterval n → BinaryFraction × ℕ)
(footprint : ℕ → Fin M × TargetInterval n → ℕ)
(eta kappa : BinaryFraction) (band : List (Fin M → TargetInterval n)) (G j t : ℕ)
(h : ∀ s coordinate, footprint s coordinate ≤ (read s coordinate).2) :
countedPreparedActionKernelAllocation mass read footprint eta kappa band G j t ≤
2 * (countedPreparedActionReadWithWork mass read eta kappa band G j t).2 := by
let coordinates := (coordinateVectorWithWork band).1
have hv := countedBandVerticesAllocation_le_work coordinates mass band
have hl := vectorMapAllocation_le_work _
(fun i => onlineLinearReadWithWork (fun child => mass.get child) read eta kappa t (coordinates.get i))
(fun i => onlineLinearReadAllocation (fun child => mass.get child) read footprint eta kappa t (coordinates.get i))
(fun i => onlineLinearReadAllocation_le_work _ _ _ _ _ _ _ (fun s _ => h s _))
have hs := selectionAllocation_le_work (countedBandVerticesWithWork coordinates mass band).1
(vectorMapWithWork _ fun i => onlineLinearReadWithWork (fun child => mass.get child)
read eta kappa t (coordinates.get i)).1 G j
unfold countedPreparedActionKernelAllocation countedPreparedActionReadWithWork
dsimp only [coordinates] at hv hl hs ⊢
omega
end EditApproximation.BinaryFraction
end OAI
|