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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 | import OAI.Combinatorics.EditApproximation.Coarse.CoarseAccessAllocation
import OAI.Combinatorics.EditApproximation.Coarse.CoarseChildAllocation
import OAI.Combinatorics.EditApproximation.Coarse.CoarseIndexedExecution
namespace OAI
namespace EditApproximation
namespace BinaryFraction
/-- Fresh bit and list cells in the actual indexed coarse evaluator.
Input arrays and random draws are shared; virtual source trees are never
materialized. Child offsets/descriptors follow the repeated reads charged
by the executable evaluator. -/
def coarseIndexedAllocation {B : ℕ} (k q L n : ℕ) (source target : List ℕ)
(two : BinaryFraction) : (d offset : ℕ) → CoarseThresholdDraws B d →
BinaryFraction → ℕ → ℕ
| 0, offset, _, allowance, start =>
let descriptor := coarseDescriptorWithWork B 0 n offset
let size := bitWordValue descriptor.1
let prune := leWithWork (nat size) allowance
let setup := coarseDescriptorAllocation B 0 n offset + allowance.ltAllocation (nat size) +
Nat.size size + 1
if prune.1 then setup else
let letter := coarsePaddedReadWithWork n source offset
setup + Nat.size n + Nat.size offset +
arithmeticMapAllocation (fun shift : Fin (2 * k + 1) =>
let mismatch := coarseReadShiftWithWork k n target letter.1 start shift
coarseReadShiftAllocation k n letter.1 start shift +
coarseGridNatAllocation allowance.denominator allowance.denominator_pos mismatch.1)
(List.finRange (2 * k + 1)) + 2 * (2 * k + 1)
| d + 1, offset, (draws, below), allowance, start =>
let descriptor := coarseDescriptorWithWork B (d + 1) n offset
let size := bitWordValue descriptor.1
let prune := leWithWork (nat size) allowance
let setup := coarseDescriptorAllocation B (d + 1) n offset + allowance.ltAllocation (nat size) +
Nat.size size + 1
if prune.1 then setup else
let capacity := bitPowerWithWork B.bits d
let childOffset := fun i : Fin B =>
coarseChildOffsetWithWork offset (bitWordValue capacity.1) i.val
let childSize := fun i : Fin B =>
coarseDescriptorWithWork B d n (bitWordValue (childOffset i).1)
let evaluate := fun (i : Fin B) a s =>
let evaluated := coarseIndexedWithWork k q L n source target two d
(bitWordValue (childOffset i).1) (below i) a s
let next := binaryNaturalAddWithWork s (bitWordValue (childSize i).1)
(evaluated.1, evaluated.2 + 3 * ((childOffset i).2 + (childSize i).2 + next.2 + 3))
let allocation := fun (i : Fin B) a s =>
coarseIndexedAllocation k q L n source target two d
(bitWordValue (childOffset i).1) (below i) a s +
3 * (coarseChildOffsetAllocation offset (bitWordValue capacity.1) i.val +
coarseDescriptorAllocation B d n (bitWordValue (childOffset i).1) +
bitAddAllocation s.bits (bitWordValue (childSize i).1).bits false)
setup + bitPowerAllocation B.bits d +
coarseIndexChildrenAllocation q L two (fun i => bitWordValue (childSize i).1)
evaluate allocation draws allowance (List.finRange B) start + B
theorem coarseIndexedAllocation_le_work {B : ℕ} (k q L n : ℕ) (source target : List ℕ)
(two : BinaryFraction) (d offset : ℕ) (draws : CoarseThresholdDraws B d)
(allowance : BinaryFraction) (start : ℕ) :
coarseIndexedAllocation k q L n source target two d offset draws allowance start ≤
2 * (coarseIndexedWithWork k q L n source target two d offset draws allowance start).2 := by
induction d generalizing offset allowance start with
| zero =>
have hd := coarseDescriptorAllocation_le_work B 0 n offset
have hp := ltAllocation_le_work allowance (nat (bitWordValue (coarseDescriptorWithWork B 0 n offset).1))
simp only [coarseIndexedAllocation, coarseIndexedWithWork]
split_ifs
· simp only [leWithWork]
omega
· have hmap := arithmeticMapAllocation_le_work
(fun shift : Fin (2 * k + 1) =>
let mismatch := coarseReadShiftWithWork k n target (coarsePaddedReadWithWork n source offset).1 start shift
let value := coarseGridNatWithWork allowance.denominator allowance.denominator_pos mismatch.1
(value.1, value.2 + mismatch.2))
(fun shift : Fin (2 * k + 1) =>
let mismatch := coarseReadShiftWithWork k n target (coarsePaddedReadWithWork n source offset).1 start shift
coarseReadShiftAllocation k n (coarsePaddedReadWithWork n source offset).1 start shift +
coarseGridNatAllocation allowance.denominator allowance.denominator_pos mismatch.1)
(List.finRange (2 * k + 1)) (by
intro shift _
have hs := coarseReadShiftAllocation_le_work k n target
(coarsePaddedReadWithWork n source offset).1 start shift
have hg := coarseGridNatAllocation_le_work allowance.denominator allowance.denominator_pos
(coarseReadShiftWithWork k n target (coarsePaddedReadWithWork n source offset).1 start shift).1
dsimp only
omega)
simp only [leWithWork, coarsePaddedReadWithWork]
simp only [coarsePaddedReadWithWork] at hmap
omega
| succ d ih =>
obtain ⟨draw, below⟩ := draws
let capacity := bitPowerWithWork B.bits d
let childOffset := fun i : Fin B => coarseChildOffsetWithWork offset (bitWordValue capacity.1) i.val
let childSize := fun i : Fin B => coarseDescriptorWithWork B d n (bitWordValue (childOffset i).1)
let evaluate := fun (i : Fin B) a s =>
let evaluated := coarseIndexedWithWork k q L n source target two d
(bitWordValue (childOffset i).1) (below i) a s
let next := binaryNaturalAddWithWork s (bitWordValue (childSize i).1)
(evaluated.1, evaluated.2 + 3 * ((childOffset i).2 + (childSize i).2 + next.2 + 3))
let allocation := fun (i : Fin B) a s =>
coarseIndexedAllocation k q L n source target two d
(bitWordValue (childOffset i).1) (below i) a s +
3 * (coarseChildOffsetAllocation offset (bitWordValue capacity.1) i.val +
coarseDescriptorAllocation B d n (bitWordValue (childOffset i).1) +
bitAddAllocation s.bits (bitWordValue (childSize i).1).bits false)
have he : ∀ i a s, allocation i a s ≤ 2 * (evaluate i a s).2 := by
intro i a s
have hr := ih (bitWordValue (childOffset i).1) (below i) a s
have ho := coarseChildOffsetAllocation_le_work offset (bitWordValue capacity.1) i.val
have hs := coarseDescriptorAllocation_le_work B d n (bitWordValue (childOffset i).1)
have ha := bitAddAllocation_le_work s.bits (bitWordValue (childSize i).1).bits false
change bitAddAllocation s.bits (bitWordValue (childSize i).1).bits false ≤
(binaryNaturalAddWithWork s (bitWordValue (childSize i).1)).2 at ha
change coarseChildOffsetAllocation offset (bitWordValue capacity.1) i.val ≤ (childOffset i).2 at ho
change coarseDescriptorAllocation B d n (bitWordValue (childOffset i).1) ≤ (childSize i).2 at hs
dsimp only [allocation, evaluate]
omega
have hc := coarseIndexChildrenAllocation_le_work k q L two (fun i => bitWordValue (childSize i).1)
evaluate allocation he draw allowance (List.finRange B) start
have hd := coarseDescriptorAllocation_le_work B (d + 1) n offset
have hp := ltAllocation_le_work allowance
(nat (bitWordValue (coarseDescriptorWithWork B (d + 1) n offset).1))
have hb := bitPowerAllocation_le_work B.bits d
simp only [coarseIndexedAllocation, coarseIndexedWithWork]
split_ifs
· simp only [leWithWork]
omega
· change coarseDescriptorAllocation B (d + 1) n offset +
allowance.ltAllocation (nat (bitWordValue (coarseDescriptorWithWork B (d + 1) n offset).1)) +
Nat.size (bitWordValue (coarseDescriptorWithWork B (d + 1) n offset).1) + 1 +
bitPowerAllocation B.bits d +
coarseIndexChildrenAllocation q L two (fun i => bitWordValue (childSize i).1)
evaluate allocation draw allowance (List.finRange B) start + B ≤
2 * ((coarseDescriptorWithWork B (d + 1) n offset).2 + capacity.2 +
(coarseIndexChildren k q L two (fun i => bitWordValue (childSize i).1)
evaluate draw allowance (List.finRange B) start).2 +
(leWithWork (nat (bitWordValue (coarseDescriptorWithWork B (d + 1) n offset).1)) allowance).2 +
Nat.size (bitWordValue (coarseDescriptorWithWork B (d + 1) n offset).1) + B + 1)
change bitPowerAllocation B.bits d ≤ capacity.2 at hb
simp only [leWithWork]
omega
end BinaryFraction
end EditApproximation
end OAI
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