File size: 4,834 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 91 92 93 94 95 96 97 98 99 | import OAI.Combinatorics.EditApproximation.Binary.BinaryGridSpace
import OAI.Combinatorics.EditApproximation.Queries.QueryBufferSpace
import OAI.Combinatorics.EditApproximation.Definitions
namespace OAI
namespace EditApproximation.BinaryFraction
theorem cellCoordinatesAllocation_le_work (side : BinaryFraction) (p radius : ℕ) :
cellCoordinatesAllocation side p radius ≤ (cellCoordinatesWithWork side p radius).2 := by
let lower := (saturatingSubtractWithWork p radius).1
let upper := (binaryNaturalAddWithWork p radius).1
let firstQ := (divWithWork (nat (bitWordValue lower)) side).1
let lastQ := (divWithWork (nat (bitWordValue upper)) side).1
let first := (naturalFloorWithWork firstQ).1
let last := (naturalFloorWithWork lastQ).1
let stop := (binaryNaturalAddWithWork (bitWordValue last) 1).1
have h1 := saturatingSubtractAllocation_le_work p radius
have h2 := binaryNaturalAddAllocation_le_work p radius
have h3 := divAllocation_le_work (nat (bitWordValue lower)) side
have h4 := divAllocation_le_work (nat (bitWordValue upper)) side
have h5 := naturalFloorAllocation_le_work firstQ
have h6 := naturalFloorAllocation_le_work lastQ
have h7 := binaryNaturalAddAllocation_le_work (bitWordValue last) 1
have h8 := saturatingSubtractAllocation_le_work (bitWordValue stop) (bitWordValue first)
unfold cellCoordinatesAllocation cellCoordinatesWithWork
dsimp only [lower, upper, firstQ, lastQ, first, last, stop] at h3 h4 h5 h6 h7 h8 ⊢
rw [List.length_range']
omega
theorem targetCellsAllocation_le_work {n : ℕ} (side : BinaryFraction) (radius : ℕ) (q : TargetInterval n) :
targetCellsAllocation side radius q ≤ (targetCellsWithWork side radius q).2 := by
have hl := cellCoordinatesAllocation_le_work side q.lo radius
have hr := cellCoordinatesAllocation_le_work side q.hi radius
have hm := arithmeticFlatMapAllocation_le_work
(fun lo => arithmeticMapWithWork (fun hi => ((lo, hi), 1)) (cellCoordinatesWithWork side q.hi radius).1)
(fun _ => arithmeticMapAllocation (fun _ => 1) (cellCoordinatesWithWork side q.hi radius).1)
(cellCoordinatesWithWork side q.lo radius).1 (fun _ _ =>
arithmeticMapAllocation_le_work _ _ _ (fun _ _ => le_rfl))
unfold targetCellsAllocation targetCellsWithWork
dsimp only
omega
theorem cellGridPointsAllocation_le_work (side : BinaryFraction) (g cell : ℕ) :
cellGridPointsAllocation side g cell ≤ (cellGridPointsWithWork side g cell).2 := by
let next := (binaryNaturalAddWithWork cell 1).1
let lower := (canonicalMulWithWork (nat cell) side).1
let upper := (canonicalMulWithWork (nat (bitWordValue next)) side).1
let firstQ := (divWithWork lower (nat g)).1
let lastQ := (divWithWork upper (nat g)).1
let first := (naturalCeilingWithWork firstQ).1
let last := (naturalCeilingWithWork lastQ).1
let count := (saturatingSubtractWithWork (bitWordValue last) (bitWordValue first)).1
let indices := List.range' (bitWordValue first) (bitWordValue count)
have h1 := binaryNaturalAddAllocation_le_work cell 1
have h2 := canonicalMulAllocation_le_work (nat cell) side
have h3 := canonicalMulAllocation_le_work (nat (bitWordValue next)) side
have h4 := divAllocation_le_work lower (nat g)
have h5 := divAllocation_le_work upper (nat g)
have h6 := naturalCeilingAllocation_le_work firstQ
have h7 := naturalCeilingAllocation_le_work lastQ
have h8 := saturatingSubtractAllocation_le_work (bitWordValue last) (bitWordValue first)
have h9 := arithmeticMapAllocation_le_work (fun i =>
(bitWordValue (binaryNaturalMulWithWork g i).1, (binaryNaturalMulWithWork g i).2 + Nat.size i + 2))
(fun i => binaryNaturalMulAllocation g i + Nat.size i) indices (by
intro i _
have h := binaryNaturalMulAllocation_le_work g i
dsimp only
omega)
unfold cellGridPointsAllocation cellGridPointsWithWork
dsimp only [next, lower, upper, firstQ, lastQ, first, last, count, indices] at h3 h4 h5 h6 h7 h8 h9 ⊢
omega
theorem cellStatesAllocation_le_work (n g : ℕ) (side : BinaryFraction) (cell : ℕ × ℕ) :
cellStatesAllocation n g side cell ≤ (cellStatesWithWork n g side cell).2 := by
have hl := cellGridPointsAllocation_le_work side g cell.1
have hr := cellGridPointsAllocation_le_work side g cell.2
have hm := arithmeticFlatMapAllocation_le_work
(fun lo => filterMapWithWork (makeStateWithWork n lo) (cellGridPointsWithWork side g cell.2).1)
(fun lo => filterMapAllocation (makeStateWithWork n lo) (makeStateAllocation n lo) (cellGridPointsWithWork side g cell.2).1)
(cellGridPointsWithWork side g cell.1).1 (fun lo _ => filterMapAllocation_le_work _ _ _
(fun hi _ => makeStateAllocation_le_work n lo hi))
unfold cellStatesAllocation cellStatesWithWork
dsimp only
omega
end EditApproximation.BinaryFraction
end OAI
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