File size: 5,979 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 | import OAI.Combinatorics.EditApproximation.Coarse.CoarseArrayControlSpace
import OAI.Combinatorics.EditApproximation.Coarse.CoarseIndexedChildren
namespace OAI
namespace EditApproximation
namespace BinaryFraction
/-- Invocation count follows the same child sequence, thresholds, recursive calls,
penalty sweeps and accumulated array as the indexed evaluator. -/
def coarseIndexChildrenInvocations {B : ℕ} (q L : ℕ) (two : BinaryFraction)
(width : Fin B → ℕ)
(evaluate : Fin B → BinaryFraction → ℕ → CoarseBinaryArray × ℕ)
(allocation : Fin B → BinaryFraction → ℕ → ℕ)
(draw : Fin B → Fin B) (allowance : BinaryFraction) : List (Fin B) → ℕ → ℕ
| [], _ => 1
| i :: rest, start =>
let threshold := coarseChildThresholdWithWork B q allowance (draw i)
let childAllowance := coarseChildAllowanceWithWork L threshold.1
let evaluated := evaluate i childAllowance.1 start
let penalty := coarseBinaryPenaltyWithWork two evaluated.1
let scale := coarseChildAllowanceWithWork q allowance
let contribution := coarseBinaryMapWithWork (coarseContributionWithWork threshold.1 scale.1) penalty.1
let remainder := coarseIndexChildren 0 q L two width evaluate draw allowance rest (start + width i)
coarseChildThresholdInvocations B q allowance (draw i) +
coarseChildAllowanceInvocations L threshold.1 + allocation i childAllowance.1 start +
coarseBinaryPenaltyInvocations two evaluated.1 + coarseChildAllowanceInvocations q allowance +
coarseBinaryMapInvocations (coarseContributionInvocations threshold.1 scale.1) penalty.1 +
coarseIndexChildrenInvocations q L two width evaluate allocation draw allowance rest (start + width i) +
coarseBinaryAddInvocations contribution.1 remainder.1 + 1
/-- The loop's first argument does not contribute to its runtime control space. -/
private theorem coarseIndexChildren_unused_k {B : ℕ} (k q L : ℕ) (two : BinaryFraction)
(width : Fin B → ℕ) (evaluate : Fin B → BinaryFraction → ℕ → CoarseBinaryArray × ℕ)
(draw : Fin B → Fin B) (allowance : BinaryFraction) (indices : List (Fin B)) (start : ℕ) :
coarseIndexChildren k q L two width evaluate draw allowance indices start =
coarseIndexChildren 0 q L two width evaluate draw allowance indices start := by
induction indices generalizing start with
| nil => rfl
| cons i rest ih => simp only [coarseIndexChildren, ih]
theorem coarseIndexChildrenInvocations_le_work {B : ℕ} (k q L : ℕ) (two : BinaryFraction)
(width : Fin B → ℕ)
(evaluate : Fin B → BinaryFraction → ℕ → CoarseBinaryArray × ℕ)
(allocation : Fin B → BinaryFraction → ℕ → ℕ)
(hallocation : ∀ i a start, allocation i a start ≤ 2 * (evaluate i a start).2 + 1)
(draw : Fin B → Fin B) (allowance : BinaryFraction) (indices : List (Fin B)) (start : ℕ) :
coarseIndexChildrenInvocations q L two width evaluate allocation draw allowance indices start ≤
2 * (coarseIndexChildren k q L two width evaluate draw allowance indices start).2 + 1 := by
rw [coarseIndexChildren_unused_k]
induction indices generalizing start with
| nil => exact Nat.le_refl 1
| cons i rest ih =>
let threshold := coarseChildThresholdWithWork B q allowance (draw i)
let childAllowance := coarseChildAllowanceWithWork L threshold.1
let evaluated := evaluate i childAllowance.1 start
let penalty := coarseBinaryPenaltyWithWork two evaluated.1
let scale := coarseChildAllowanceWithWork q allowance
let contribution := coarseBinaryMapWithWork (coarseContributionWithWork threshold.1 scale.1) penalty.1
let remainder := coarseIndexChildren 0 q L two width evaluate draw allowance rest (start + width i)
have ht := coarseChildThresholdInvocations_le_work B q allowance (draw i)
have ha := coarseChildAllowanceInvocations_le_work L threshold.1
have he := hallocation i childAllowance.1 start
have hp := coarseBinaryPenaltyInvocations_le_work two evaluated.1
have hs := coarseChildAllowanceInvocations_le_work q allowance
have hm := coarseBinaryMapInvocations_le_work
(coarseContributionWithWork threshold.1 scale.1)
(coarseContributionInvocations threshold.1 scale.1)
(coarseContributionInvocations_le_work threshold.1 scale.1) penalty.1
have hr := ih (start + width i)
have had := coarseBinaryAddInvocations_le_work contribution.1 remainder.1
change coarseChildThresholdInvocations B q allowance (draw i) +
coarseChildAllowanceInvocations L threshold.1 + allocation i childAllowance.1 start +
coarseBinaryPenaltyInvocations two evaluated.1 + coarseChildAllowanceInvocations q allowance +
coarseBinaryMapInvocations (coarseContributionInvocations threshold.1 scale.1) penalty.1 +
coarseIndexChildrenInvocations q L two width evaluate allocation draw allowance rest (start + width i) +
coarseBinaryAddInvocations contribution.1 remainder.1 + 1 ≤
2 * (threshold.2 + childAllowance.2 + evaluated.2 + penalty.2 + scale.2 + contribution.2 +
remainder.2 + (coarseBinaryAddWithWork contribution.1 remainder.1).2 +
Nat.size start + Nat.size (width i) + 9) + 1
change coarseChildThresholdInvocations B q allowance (draw i) ≤ threshold.2 at ht
change coarseChildAllowanceInvocations L threshold.1 ≤ childAllowance.2 at ha
change allocation i childAllowance.1 start ≤ 2 * evaluated.2 + 1 at he
change coarseBinaryPenaltyInvocations two evaluated.1 ≤ 2 * penalty.2 + 9 at hp
change coarseChildAllowanceInvocations q allowance ≤ scale.2 at hs
change coarseBinaryMapInvocations (coarseContributionInvocations threshold.1 scale.1) penalty.1 ≤
contribution.2 + 2 at hm
change coarseIndexChildrenInvocations q L two width evaluate allocation draw allowance rest (start + width i) ≤
2 * remainder.2 + 1 at hr
omega
end BinaryFraction
end EditApproximation
end OAI
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