File size: 4,476 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 | import OAI.Combinatorics.EditApproximation.Queries.QueryFamilyRefinement
import OAI.Combinatorics.EditApproximation.Queries.QueryPhysicalProgramCost
import OAI.Combinatorics.EditApproximation.Queries.QueryFamilyReaderBounds
namespace OAI
namespace EditApproximation
section Family
variable {α : Type*} {M J R : ℕ} [NeZero M] (source target : List α)
(N P F exponent H Q eM eF B E Scale : ℕ) (hP : 0 < P) (hM : M = 2 ^ eM) (hF : F = 2 ^ eF)
(tau a eta kappa delta : BinaryFraction) (passes copies pass copy T S t : ℕ) (ht : 0 < t)
(node : PhysicalNode M J) (hbelow : node.1.val < J) (time : Fin R) (q : TargetInterval target.length)
(oracle : Sum (PhysicalFamilyScalarKey (M := M) (J := J) (R := R)
target N P F exponent H Q passes copies)
{child : PhysicalTableRequest M J target.length // child.combinedRank T S <
(PhysicalTableRequest.refinement pass copy node T t q).combinedRank T S} → BinaryFraction)
(readWork : Sum (PhysicalFamilyScalarKey (M := M) (J := J) (R := R)
target N P F exponent H Q passes copies)
{child : PhysicalTableRequest M J target.length // child.combinedRank T S <
(PhysicalTableRequest.refinement pass copy node T t q).combinedRank T S} → ℕ)
local notation "tableOracle" => (fun key => oracle (Sum.inr key))
open BinaryFraction
theorem queryFamilyRefinement_work_bound (V C : ℕ)
(hn : PreparedKernelNumericBounds (nx := source.length) (ny := target.length) (M := M)
N P F exponent H Q eM eF B E Scale (t - 1) tau a eta kappa delta (fun _ _ => 0))
(htime : Nat.size t ≤ B) (hPd : ∃ j : ℕ, P = 2 ^ j)
(hV : 2 ≤ V) (hVB : V ≤ B) (hMean : warmupAverageWordBudget T V ≤ B)
(hc : ∀ key, key.val.pass = pass → key.val.copy = copy → readWork (Sum.inr key) ≤ C)
(hs : ∀ key, readWork (Sum.inl key) ≤ C)
(hv : ∀ key, (tableOracle key).width ≤ V)
(hsv : ∀ key, (oracle (Sum.inl key)).width ≤ B)
(hnonneg : ∀ key, 0 ≤ (tableOracle key).value)
(hprior : ∀ input : Fin M × ℕ × TargetInterval target.length,
((queryPhysicalEarlierRead copies pass copy T S t node hbelow q input).value tableOracle).value ≤
((queryPhysicalWarmupRead copies pass copy T S t ht node q
(physicalChild node hbelow input.1) (by change node.1.val ≤ node.1.val + 1; omega)
0 (Nat.zero_le T) input.2.2).value tableOracle).value)
(hp : Nat.size pass ≤ B) (hcopy : Nat.size copy ≤ B)
(hps : Nat.size (passes + 1) ≤ B) (hcs : Nat.size (copies + 1) ≤ B)
(hbound : Nat.size (localOnlineMassBound M P H) ≤ B) :
(queryFamilyRefinement source target N P F exponent H Q eM eF hP hM hF tau a eta kappa delta
passes copies pass copy T S t ht node hbelow time q).work oracle readWork ≤
queryLocalProgramBitBudget M N P F exponent eM eF B Scale t
(queryPhysicalDataBudget T J B V C (queryFamilySourceReadBudget J B (C + 1))) := by
have hB : 2 ≤ B := by have := hn.largeBits; omega
have htarget : Nat.size target.length ≤ B := by have := hn.targetSize; omega
unfold queryFamilyRefinement
apply queryPhysicalRandomDataRequest_work_bound source target N P F exponent H Q eM eF B E Scale
hP hM hF tau a eta kappa delta copies pass copy T S t ht node hbelow _ _ q oracle readWork
V C (queryFamilySourceReadBudget J B (C + 1)) hn htime hPd hV hVB hMean hc hv hnonneg hprior
· intro key
exact queryFamilyGroupRead_width target N P F exponent H Q passes copies pass copy B
(fun key => oracle (Sum.inl key)) hB hsv ⟨node, hbelow⟩ time key
· intro key range
exact queryFamilyOnlineRead_width target N P F exponent H Q passes copies pass copy B
(fun key => oracle (Sum.inl key)) hB hsv ⟨node, hbelow⟩ time key range
· intro key hk
exact queryFamilyGroupRead_bounded_work target N P F exponent H Q passes copies pass copy B (C + 1)
(fun key => oracle (Sum.inl key)) (fun key => 1 + readWork (Sum.inl key))
hn.inputSize hp hcopy hps hcs (fun key => by have := hs key; omega)
htarget hn.thetaSize ⟨node, hbelow⟩ time key hk
· intro key range hk hr
exact queryFamilyOnlineRead_bounded_work target N P F exponent H Q passes copies pass copy B (C + 1)
(fun key => oracle (Sum.inl key)) (fun key => 1 + readWork (Sum.inl key))
hn.inputSize hp hcopy hps hcs (fun key => by have := hs key; omega)
hbound ⟨node, hbelow⟩ time key range hk hr
end Family
end EditApproximation
end OAI
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