File size: 3,671 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 | import OAI.Combinatorics.EditApproximation.Refinement.ComputedPhysicalCoarseWork
import OAI.Combinatorics.EditApproximation.Refinement.ComputedExecutionLaw
import OAI.Combinatorics.EditApproximation.Probability.FiniteProductWork
import OAI.Combinatorics.EditApproximation.Definitions
namespace OAI
namespace EditApproximation
open MeasureTheory Filter
section Symbolic
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)
theorem symbolicComputedExecutionCoarseWork_expected_le (budget : ℝ)
(hbudget : SymbolicComputedCoarseBudget (B := B) source target N depth encode budget) :
(∫ execution, symbolicComputedExecutionCoarseWork source target N depth encode execution
∂physicalCoarseExecutionLaw (R := R) source target N (integerParameters N).P
(inputHeight N) B depth) ≤ budget := by
exact rejectionProductWork_expected_le _ _
(fun outcome => symbolicComputedCoarseWork source target N depth encode outcome.2 outcome.1)
budget hbudget
theorem symbolicComputedExecutionCoarseWork_integrable :
Integrable (symbolicComputedExecutionCoarseWork source target N depth encode)
(physicalCoarseExecutionLaw (R := R) source target N (integerParameters N).P
(inputHeight N) B depth) := by
let μ := coarseFullSeedMeasure B depth
(fun entry => physicalSource source entry.1)
(fun entry => substring target entry.2.lo entry.2.hi)
let range := physicalFamilyRange (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)
let f := fun outcome : PhysicalCoarseDraw source target B depth × (∀ key, Fin (range key)) =>
symbolicComputedCoarseWork source target N depth encode outcome.2 outcome.1
have hf : Integrable f ((μ.prod (rejectionRuntimeArrayLaw range)).map
(rejectionProductAccepted range)) := by
rw [rejectionProductAccepted_law]
exact Integrable.of_finite
exact hf.comp_measurable (rejectionProductAccepted_measurable range)
end Symbolic
theorem computedExecutionCoarseWork_integrable (source target : List ℕ) (N : ℕ) :
Integrable (computedExecutionCoarseWork source target N) (computedExecutionLaw source target N) :=
symbolicComputedExecutionCoarseWork_integrable source target N
(computedPhysicalCoarseDepth source target N) (computedFamilyMemoCode source target N (inputHeight N ^ 2))
theorem computedExecutionCoarseWork_expected_near_linear (epsilon : ℝ) (hepsilon : 0 < epsilon) :
∀ᶠ N in atTop, ∀ source target : List ℕ, source.length + target.length ≤ N →
(∫ execution, computedExecutionCoarseWork source target N execution
∂computedExecutionLaw source target N) ≤ (N + 2 : ℝ) ^ (1 + epsilon) := by
filter_upwards [computedPhysicalCoarseWork_expected_near_linear epsilon hepsilon] with N hN
intro source target hsize
exact symbolicComputedExecutionCoarseWork_expected_le source target N
(computedPhysicalCoarseDepth source target N) (computedFamilyMemoCode source target N (inputHeight N ^ 2))
_ (hN source target hsize)
end EditApproximation
end OAI
|