File size: 2,950 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 75 76 77 78 79 80 | import OAI.Combinatorics.EditApproximation.Queries.QueryFamilyRefinementCost
import OAI.Combinatorics.EditApproximation.Binary.BinaryPowerStorage
import OAI.Combinatorics.EditApproximation.Probability.FiniteBitPolynomial
namespace OAI
namespace EditApproximation
private abbrev GP := FiniteBitPolynomial (Fin 6)
private def gp_storedSumWordBudget (n B : GP) : GP :=
(B + 1) + FiniteBitPolynomial.bitLength n + 2 * (B + 2) + 2 * (1 + n * ((B + 1) + (B + 1))) + 1
private def gp_storedSumBitBudget (n B : GP) : GP :=
1 + n * (100000 * (gp_storedSumWordBudget n B + 1) ^ 2)
private def gp_bandMassTotalStorageBudget (M B : GP) : GP :=
FiniteBitPolynomial.bitLength M + B + 2 * (1 + M * (B + 1)) + 2
private def gp_warmupAverageBitBudget (n B : GP) : GP :=
gp_storedSumBitBudget n B + 6 * n +
2100 * (gp_bandMassTotalStorageBudget n B + FiniteBitPolynomial.bitLength n + 4) ^ 2 + FiniteBitPolynomial.bitLength n + 2
private def gp_queryPhysicalDataBudget (T J B V C W : GP) : GP :=
W + (T + 1 + T * (7 * J + 30 + (C + 1) + 3) + gp_warmupAverageBitBudget T V) +
(8 * B + 7 * J + 32 + (C + 1)) + 1
private def gp_queryFamilySourceReadBudget (J B W : GP) : GP :=
200 * (1000 * (B + 1) + 1000 + 2) ^ 2 + 120 * (B + 1) ^ 2 + 10 * J + 100 + W
private def gp_bitPowerBitBudget (n B : GP) : GP :=
1 + n * (200 * ((n + 1) * (B + 1) + 1) ^ 2)
private def gp_powerDenominatorBitBudget (n A B : GP) : GP :=
gp_bitPowerBitBudget n B + 5000 * (A + (n * B + 1) + 2) ^ 2 + 1
private def callbackPolynomial : GP :=
gp_queryPhysicalDataBudget (.input 0) (.input 1) (.input 2) (.input 3) (.input 4)
(gp_queryFamilySourceReadBudget (.input 1) (.input 2) (.input 5))
def queryCallbackShellDegree : ℕ := callbackPolynomial.degree
theorem queryCallbackShell_fixedPower (T J B V C W X : ℕ) (hX : 2 ≤ X)
(hT : T ≤ X) (hJ : J ≤ X) (hB : B ≤ X) (hV : V ≤ X) (hC : C ≤ X) (hW : W ≤ X) :
BinaryFraction.queryPhysicalDataBudget T J B V C (BinaryFraction.queryFamilySourceReadBudget J B W) ≤
X ^ queryCallbackShellDegree := by
have h := callbackPolynomial.eval_le_pow ![T, J, B, V, C, W] X hX (by
intro i
fin_cases i
· exact hT
· exact hJ
· exact hB
· exact hV
· exact hC
· exact hW)
exact h
private def normalizationPolynomial : GP :=
gp_powerDenominatorBitBudget (.input 0) (.input 1) (.input 2)
def queryNormalizationBitDegree : ℕ := normalizationPolynomial.degree
theorem queryNormalization_fixedPower (n A B X : ℕ) (hX : 2 ≤ X)
(hn : n ≤ X) (hA : A ≤ X) (hB : B ≤ X) :
BinaryFraction.powerDenominatorBitBudget n A B ≤ X ^ queryNormalizationBitDegree := by
have h := normalizationPolynomial.eval_le_pow ![n, A, B, 0, 0, 0] X hX (by
intro i
fin_cases i
· exact hn
· exact hA
· exact hB
· exact Nat.zero_le _
· exact Nat.zero_le _
· exact Nat.zero_le _)
exact h
end EditApproximation
end OAI
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