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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 | import OAI.Combinatorics.EditApproximation.Coarse.CoarseAccessControlSpace
import OAI.Combinatorics.EditApproximation.Coarse.CoarseChildControlSpace
import OAI.Combinatorics.EditApproximation.Coarse.CoarseIndexedExecution
namespace OAI
namespace EditApproximation
namespace BinaryFraction
/-- Conservative recursive invocation count for the indexed evaluator,
including scalar callees, index traversals and repeated descriptor reads. -/
def coarseIndexedInvocations {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 := coarseDescriptorInvocations B 0 n offset + compareInvocations allowance (nat size) + 2
if prune.1 then setup else
let letter := coarsePaddedReadWithWork n source offset
setup + coarsePaddedReadInvocations n offset +
arithmeticMapInvocations (fun shift : Fin (2 * k + 1) =>
let mismatch := coarseReadShiftWithWork k n target letter.1 start shift
coarseReadShiftInvocations k n target letter.1 start shift +
coarseGridNatInvocations allowance.denominator allowance.denominator_pos mismatch.1 + 1)
(List.finRange (2 * k + 1)) + (2 * k + 1) + 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 := coarseDescriptorInvocations B (d + 1) n offset + compareInvocations allowance (nat size) + 2
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 =>
coarseIndexedInvocations k q L n source target two d
(bitWordValue (childOffset i).1) (below i) a s +
3 * (coarseChildOffsetInvocations offset (bitWordValue capacity.1) i.val +
coarseDescriptorInvocations B d n (bitWordValue (childOffset i).1) +
bitAddInvocations s.bits (bitWordValue (childSize i).1).bits false) + 1
setup + bitPowerInvocations B.bits d +
coarseIndexChildrenInvocations q L two (fun i => bitWordValue (childSize i).1)
evaluate allocation draws allowance (List.finRange B) start + B + 1
theorem arithmeticMapInvocations_with_indices_le_work {α β : Type*} (f : α → β × ℕ)
(invocations : α → ℕ) (values : List α)
(h : ∀ a ∈ values, invocations a ≤ 2 * (f a).2 + 1) :
arithmeticMapInvocations invocations values + values.length + 1 ≤
2 * (arithmeticMapWithWork f values).2 + 2 := by
induction values with
| nil => exact Nat.le_refl 2
| cons a rest ih =>
have hh := h a List.mem_cons_self
have ht := ih (fun b hb => h b (List.mem_cons_of_mem _ hb))
simp only [arithmeticMapInvocations, arithmeticMapWithWork, List.length_cons]
omega
theorem coarseIndexedInvocations_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) :
coarseIndexedInvocations 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 + 1 := by
induction d generalizing offset allowance start with
| zero =>
have hd := coarseDescriptorInvocations_le_work B 0 n offset
have hp := compareInvocations_le_work allowance (nat (bitWordValue (coarseDescriptorWithWork B 0 n offset).1))
simp only [coarseIndexedInvocations, coarseIndexedWithWork]
split_ifs
· simp only [leWithWork]
omega
· have hmap := arithmeticMapInvocations_with_indices_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
coarseReadShiftInvocations k n target (coarsePaddedReadWithWork n source offset).1 start shift +
coarseGridNatInvocations allowance.denominator allowance.denominator_pos mismatch.1 + 1)
(List.finRange (2 * k + 1)) (by
intro shift _
have hs := coarseReadShiftInvocations_le_work k n target
(coarsePaddedReadWithWork n source offset).1 start shift
have hg := coarseGridNatInvocations_le_work allowance.denominator allowance.denominator_pos
(coarseReadShiftWithWork k n target (coarsePaddedReadWithWork n source offset).1 start shift).1
dsimp only
omega)
have hpadded := coarsePaddedReadInvocations_le_work n source offset
simp only [List.length_finRange] at hmap
simp only [leWithWork, coarsePaddedReadWithWork] at hpadded ⊢
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 =>
coarseIndexedInvocations k q L n source target two d
(bitWordValue (childOffset i).1) (below i) a s +
3 * (coarseChildOffsetInvocations offset (bitWordValue capacity.1) i.val +
coarseDescriptorInvocations B d n (bitWordValue (childOffset i).1) +
bitAddInvocations s.bits (bitWordValue (childSize i).1).bits false) + 1
have he : ∀ i a s, allocation i a s ≤ 2 * (evaluate i a s).2 + 1 := by
intro i a s
have hr := ih (bitWordValue (childOffset i).1) (below i) a s
have ho := coarseChildOffsetInvocations_le_work offset (bitWordValue capacity.1) i.val
have hs := coarseDescriptorInvocations_le_work B d n (bitWordValue (childOffset i).1)
have ha := bitAddInvocations_le_work s.bits (bitWordValue (childSize i).1).bits false
change bitAddInvocations s.bits (bitWordValue (childSize i).1).bits false ≤
(binaryNaturalAddWithWork s (bitWordValue (childSize i).1)).2 at ha
change coarseChildOffsetInvocations offset (bitWordValue capacity.1) i.val ≤ (childOffset i).2 at ho
change coarseDescriptorInvocations B d n (bitWordValue (childOffset i).1) ≤ (childSize i).2 at hs
dsimp only [allocation, evaluate]
omega
have hc := coarseIndexChildrenInvocations_le_work k q L two (fun i => bitWordValue (childSize i).1)
evaluate allocation he draw allowance (List.finRange B) start
have hd := coarseDescriptorInvocations_le_work B (d + 1) n offset
have hp := compareInvocations_le_work allowance
(nat (bitWordValue (coarseDescriptorWithWork B (d + 1) n offset).1))
have hb := bitPowerInvocations_le_work B.bits d
simp only [coarseIndexedInvocations, coarseIndexedWithWork]
split_ifs
· simp only [leWithWork]
omega
· change coarseDescriptorInvocations B (d + 1) n offset +
compareInvocations allowance (nat (bitWordValue (coarseDescriptorWithWork B (d + 1) n offset).1)) + 2 +
bitPowerInvocations B.bits d +
coarseIndexChildrenInvocations q L two (fun i => bitWordValue (childSize i).1)
evaluate allocation draw allowance (List.finRange B) start + B + 1 ≤
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) + 1
change bitPowerInvocations B.bits d ≤ capacity.2 at hb
simp only [leWithWork]
omega
end BinaryFraction
end EditApproximation
end OAI
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