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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 | import OAI.Combinatorics.EditApproximation.Refinement.ComputedPhysicalWork
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalFamilyMemoCoarse
import OAI.Combinatorics.EditApproximation.Definitions
namespace OAI
/-! Polynomial coarse-sampling budgets for the computed dimensions. -/
namespace EditApproximation
open MeasureTheory Filter
section Coarse
variable {M J R B : ℕ} [NeZero M] [NeZero R] [NeZero B]
(source target : List ℕ) (N : ℕ)
(depth : PhysicalEntry M J target.length → ℕ)
(encode : Sum (PhysicalFamilyScalarKey (M := M) (J := J) (R := R)
target N (integerParameters N).P (inputSmallLog N) (40 * smallLogExponent N)
(inputHeight N) (inputSmallLog N ^ 80) (computedSeedPassCount N) (inputHeight N ^ 2))
(ChargedPhysicalRequest M J target.length (computedSeedPassCount N + 1)
(inputHeight N ^ 2) (integerParameters N).T (integerParameters N).S) → ℕ)
def SymbolicComputedCoarseBudget (budget : ℝ) : Prop :=
∀ draw, integral (coarseFullSeedMeasure B depth
(fun entry => physicalSource source entry.1)
(fun entry => substring target entry.2.lo entry.2.hi))
(symbolicComputedCoarseWork (B := B) source target N depth encode draw) ≤ budget
theorem symbolicComputedCoarseWork_expected_le
(hN : 0 < N) (hell : 200 ≤ inputSmallLog N) (hsize : source.length + target.length ≤ N)
(hM : M = (integerParameters N).M) (hJ : J = intervalTreeDepth M source.length)
(hB : B = (integerParameters N).B)
(hdepth : ∀ entry, depth entry = computedCoarseDepth N
(coarseEntryWidth (physicalSource source entry.1) (substring target entry.2.lo entry.2.hi)))
(hinjective : Function.Injective encode)
(hbounded : ∀ key, encode key < 2 ^ (16384 * inputHeight N + 5))
(hinverse : coarseInverseAllowanceFactor B (initialThresholdQ N) (initialLambda N) ≤
(inputHeight N : ℝ) ^ 8) :
SymbolicComputedCoarseBudget (B := B) source target N depth encode
(computedWidthEnvelope N * computedCoarseWidthOverhead N) := by
intro draw
have hH : 105 ≤ inputHeight N := (by omega : 105 ≤ inputSmallLog N).trans
(computed_smallLog_le_height N)
apply physicalFamilyMemoCoarseWork_expected_le source target N (integerParameters N).P
(inputHeight N) (integerParameters N).T (integerParameters N).S (computedSeedPassCount N)
(inputHeight N ^ 2) (scheduledSeedFactor N) (scheduledSeedMultiplier N)
(rationalParameters (inputSmallLog N)).tau (rationalParameters (inputSmallLog N)).eta
(rationalParameters (inputSmallLog N)).kappa (rationalParameters (inputSmallLog N)).delta
draw B depth (computedParameterP_pos N) hN hell hB hsize hdepth hinverse
(16384 * inputHeight N + 5) encode hinjective hbounded _ (by rfl) (computedWidthEnvelope N)
intro coarse
let initial := clampedSeedNat (physicalCoarseSeedTable source target N B depth coarse)
let root := localFinalRootRequest (M := M) (J := J) target N (integerParameters N).T
(integerParameters N).S (inputHeight N) (Nat.two_pow_pos _) (finalTargetState target.length)
let tree := physicalFamilyDependencyTree source target N (integerParameters N).P
(inputHeight N) (integerParameters N).T (integerParameters N).S (computedSeedPassCount N)
(inputHeight N ^ 2) (scheduledSeedFactor N) (scheduledSeedMultiplier N)
(rationalParameters (inputSmallLog N)).tau (rationalParameters (inputSmallLog N)).eta
(rationalParameters (inputSmallLog N)).kappa (rationalParameters (inputSmallLog N)).delta
initial (computedParameterP_pos N) draw root
have hv := physicalFamilyDependencyTree_valid source target N (integerParameters N).P
(inputHeight N) (integerParameters N).T (integerParameters N).S (computedSeedPassCount N)
(inputHeight N ^ 2) (scheduledSeedFactor N) (scheduledSeedMultiplier N)
(rationalParameters (inputSmallLog N)).tau (rationalParameters (inputSmallLog N)).eta
(rationalParameters (inputSmallLog N)).kappa (rationalParameters (inputSmallLog N)).delta
initial (computedParameterP_pos N) draw hN hell hM rfl rfl rfl le_rfl
(fun pass => (scheduledSeedFactor_bounds N pass).1)
(fun pass => (scheduledSeedFactor_bounds N pass).2) root
have hd : tree.depth = intervalTreeDepth (integerParameters N).M source.length := by
rw [physicalFamilyDependencyTree_depth]
exact hJ.trans (congrArg (fun m => intervalTreeDepth m source.length) hM)
have hk := physicalFamilyDependencyTree_rank source target N (integerParameters N).P
(inputHeight N) (integerParameters N).T (integerParameters N).S (computedSeedPassCount N)
(inputHeight N ^ 2) (scheduledSeedFactor N) (scheduledSeedMultiplier N)
(rationalParameters (inputSmallLog N)).tau (rationalParameters (inputSmallLog N)).eta
(rationalParameters (inputSmallLog N)).kappa (rationalParameters (inputSmallLog N)).delta
initial (computedParameterP_pos N) draw rfl rfl rfl root
have hw := computed_widthWork_le_envelope N source.length hN (by omega) hell hH tree
(by simpa only [hM] using hv) hd hk
simpa only [hM] using hw
theorem symbolicComputedCoarseBudget_mono (a b : ℝ) (hab : a ≤ b)
(h : SymbolicComputedCoarseBudget (B := B) source target N depth encode a) :
SymbolicComputedCoarseBudget (B := B) source target N depth encode b := fun draw => (h draw).trans hab
end Coarse
end EditApproximation
end OAI
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