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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 | import OAI.Combinatorics.EditApproximation.Refinement.PhysicalCoarseSamplingWork
import OAI.Combinatorics.EditApproximation.Refinement.ComputedCoarseBitWork
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 symbolicComputedExecutionCoarseSamplingWork_integrable :
Integrable (symbolicComputedExecutionCoarseSamplingWork 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)) =>
symbolicComputedExecutionCoarseSamplingWork source target N depth encode
(outcome.1, (fun _ => 0, outcome.2))
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)
omit [NeZero B] in
theorem symbolicComputedExecutionCoarseSamplingWork_bound
(execution : PhysicalCoarseExecution (R := R) source target N
(integerParameters N).P (inputHeight N) B depth) :
symbolicComputedExecutionCoarseSamplingWork source target N depth encode execution ≤
(Nat.size B : ℝ) * symbolicComputedExecutionCoarseWork source target N depth encode execution := by
exact physicalFamilyCoarseSamplingWork_bound 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
execution.2.2 depth (16384 * inputHeight N + 5) encode
(localFinalRootRequest target N (integerParameters N).T (integerParameters N).S
(inputHeight N) (Nat.two_pow_pos _) (finalTargetState target.length)) execution.1
end Symbolic
theorem computedExecutionCoarseSamplingWork_integrable (source target : List ℕ) (N : ℕ) :
Integrable (computedExecutionCoarseSamplingWork source target N) (computedExecutionLaw source target N) :=
symbolicComputedExecutionCoarseSamplingWork_integrable source target N
(computedPhysicalCoarseDepth source target N) (computedFamilyMemoCode source target N (inputHeight N ^ 2))
theorem computedExecutionCoarseSamplingWork_bound (source target : List ℕ) (N : ℕ)
(execution : ComputedExecutionOutcome source target N) :
computedExecutionCoarseSamplingWork source target N execution ≤
(Nat.size (integerParameters N).B : ℝ) * computedExecutionCoarseWork source target N execution :=
symbolicComputedExecutionCoarseSamplingWork_bound source target N
(computedPhysicalCoarseDepth source target N) (computedFamilyMemoCode source target N (inputHeight N ^ 2)) execution
theorem computedExecutionCoarseSamplingWork_expected_near_linear (epsilon : ℝ) (hepsilon : 0 < epsilon) :
∀ᶠ N in atTop, ∀ source target : List ℕ, source.length + target.length ≤ N →
(∫ execution, computedExecutionCoarseSamplingWork source target N execution
∂computedExecutionLaw source target N) ≤ (N + 2 : ℝ) ^ (1 + epsilon) := by
have he : 0 < epsilon / 2 := half_pos hepsilon
filter_upwards [computedExecutionCoarseWork_expected_near_linear (epsilon / 2) he,
fixed_height_power_subpolynomial 2 (epsilon / 2) he,
eventually_ge_atTop 1, inputSmallLog_tendsto.eventually (eventually_ge_atTop 200)] with N hc hp hN hell
intro source target hsize
have hB := Nat.size_le_size (computed_coarse_B_upper N (by omega) hell)
rw [Nat.size_pow] at hB
have hH := computed_coarse_height_large N (by omega) hell
have hsizeB : Nat.size (integerParameters N).B ≤ inputHeight N ^ 2 := by nlinarith only [hB, hH]
have hb : (∫ execution, computedExecutionCoarseSamplingWork source target N execution
∂computedExecutionLaw source target N) ≤
(Nat.size (integerParameters N).B : ℝ) *
(∫ execution, computedExecutionCoarseWork source target N execution ∂computedExecutionLaw source target N) := by
calc
_ ≤ ∫ execution, (Nat.size (integerParameters N).B : ℝ) *
computedExecutionCoarseWork source target N execution ∂computedExecutionLaw source target N :=
integral_mono (computedExecutionCoarseSamplingWork_integrable source target N)
((computedExecutionCoarseWork_integrable source target N).const_mul _)
(computedExecutionCoarseSamplingWork_bound source target N)
_ = _ := by rw [integral_const_mul]
have hsizeBReal : (Nat.size (integerParameters N).B : ℝ) ≤ (inputHeight N : ℝ) ^ 2 := by exact_mod_cast hsizeB
calc
_ ≤ (Nat.size (integerParameters N).B : ℝ) * (N + 2 : ℝ) ^ (1 + epsilon / 2) :=
hb.trans (mul_le_mul_of_nonneg_left (hc source target hsize) (Nat.cast_nonneg _))
_ ≤ (inputHeight N : ℝ) ^ 2 * (N + 2 : ℝ) ^ (1 + epsilon / 2) :=
mul_le_mul_of_nonneg_right hsizeBReal (by positivity)
_ ≤ (N + 2 : ℝ) ^ (epsilon / 2) * (N + 2 : ℝ) ^ (1 + epsilon / 2) :=
mul_le_mul_of_nonneg_right hp (by positivity)
_ = _ := by rw [← Real.rpow_add (by positivity)]; congr 1; ring
end EditApproximation
end OAI
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