File size: 3,882 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 | import OAI.Combinatorics.EditApproximation.Refinement.ComputedExecutionLaw
import OAI.Combinatorics.EditApproximation.Refinement.ComputedRetainedValues
import OAI.Combinatorics.EditApproximation.Probability.FiniteProductWork
import OAI.Combinatorics.EditApproximation.Definitions
namespace OAI
namespace EditApproximation
open MeasureTheory Filter
theorem computedFamilyRejectionWork_integrable (source target : List ℕ) (N : ℕ)
(seed : ComputedFamilySeed source target N) :
Integrable (fun execution : ComputedRejectionRuntime source target N =>
computedFamilyRejectionWork source target N (seed, execution))
(rejectionRuntimeArrayLaw (physicalFamilyRange (M := (integerParameters N).M)
(J := intervalTreeDepth (integerParameters N).M source.length) (R := inputHeight N ^ 2)
target N (integerParameters N).P (inputSmallLog N) (40 * smallLogExponent N)
(inputHeight N) (inputSmallLog N ^ 80) (computedSeedPassCount N) (inputHeight N ^ 2))) :=
physicalFamilyMemoBits_integrable source target N (integerParameters N).P
(integerParameters N).T (integerParameters N).S (inputHeight N) (Nat.two_pow_pos _)
(16384 * inputHeight N + 5) (computedFamilyMemoCode source target N (inputHeight N ^ 2))
(finalTargetState target.length) seed
theorem computedExecutionRejectionWork_expected_near_linear (epsilon : ℝ) (hepsilon : 0 < epsilon) :
∀ᶠ N in atTop, ∀ source target : List ℕ, source.length + target.length ≤ N →
integral (computedExecutionLaw source target N) (computedExecutionRejectionWork source target N) ≤
(N + 2 : ℝ) ^ (1 + epsilon) := by
filter_upwards [computedFamilyRejectionWork_expected_near_linear epsilon hepsilon] with N hN
intro source target hsize
let incoming := physicalCoarseSeedLaw (M := (integerParameters N).M)
(J := intervalTreeDepth (integerParameters N).M source.length) source target N
(integerParameters N).B (computedPhysicalCoarseDepth source target N)
have hmap := integral_map (μ := computedExecutionLaw source target N)
(computedExecutionProjection_measurable source target N).aemeasurable
(measurable_of_countable (computedFamilyRejectionWork source target N)).aestronglyMeasurable
rw [computedExecutionProjection_law] at hmap
change (∫ execution, computedFamilyRejectionWork source target N
(computedExecutionProjection source target N execution) ∂computedExecutionLaw source target N) ≤ _
rw [← hmap]
apply finite_outer_expected_le incoming _ (computedFamilyRejectionWork source target N)
(computedFamilyRejectionWork_integrable source target N) _
intro seed
exact hN source target hsize seed
theorem computedExecution_requests_near_linear (epsilon : ℝ) (hepsilon : 0 < epsilon) :
∀ᶠ N in atTop, ∀ source target : List ℕ, source.length + target.length ≤ N →
∀ execution : ComputedExecutionOutcome source target N,
((computedFamilyResult source target N (computedExecutionAccepted source target N execution)).requests : ℝ) ≤
(N + 2 : ℝ) ^ (1 + epsilon) := by
filter_upwards [computedFamilyResult_requests_near_linear epsilon hepsilon] with N hN
intro source target hsize execution
exact hN source target hsize _
theorem computedExecution_retained_storage (source target : List ℕ) (N : ℕ)
(hN : 0 < N) (hell : 2 ^ 1000 ≤ inputSmallLog N) (hsize : source.length + target.length ≤ N)
(hB : Nat.size (localOnlineMassBound (integerParameters N).M (integerParameters N).P
(inputHeight N)) ≤ inputHeight N) (execution : ComputedExecutionOutcome source target N) :
(computedFamilyResult source target N (computedExecutionAccepted source target N execution)).memory.rationalStorage ≤
(N + 2) ^ 65556 :=
computedFamilyResult_retained_storage source target N hN hell hsize hB _
end EditApproximation
end OAI
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