File size: 3,227 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 | import OAI.Combinatorics.EditApproximation.Binary.BinaryOnlineParameters
import OAI.Combinatorics.EditApproximation.Refinement.ComputedCacheWords
namespace OAI
/-! The actual binary online constants fit the same polynomial word cap as
the normalized local cache. No input-probability or good-seed assumption is
used here. -/
namespace EditApproximation
private theorem online_parameter_width_cap (H B : ℕ) (hH : 64 ≤ H) (hB : B ≤ H) :
5 * B + 1 ≤ H ^ 2 ∧ 4 * B + 1 ≤ H ^ 2 := by
constructor <;> nlinarith
theorem computedBinaryOnlineParameters_caps (N : ℕ) (word : List Bool)
(hvalue : bitWordValue word = inputSmallLog N)
(hwidth : word.length ≤ Nat.size (inputSmallLog N)) (hH : 64 ≤ inputHeight N)
(hpositive : 0 < bitWordValue word) :
(BinaryFraction.onlineParametersWithWork word hpositive).1.1.width ≤ inputHeight N ^ 2 ∧
(BinaryFraction.onlineParametersWithWork word hpositive).1.2.width ≤ inputHeight N ^ 2 ∧
|(BinaryFraction.onlineParametersWithWork word hpositive).1.1.value| ≤
((inputHeight N ^ 5 : ℕ) : ℚ) ∧
0 ≤ (BinaryFraction.onlineParametersWithWork word hpositive).1.2.value ∧
(BinaryFraction.onlineParametersWithWork word hpositive).1.2.value ≤ 1 := by
have hsmall : Nat.size (inputSmallLog N) ≤ inputHeight N :=
(Nat.size_le.mpr Nat.lt_two_pow_self).trans (computed_smallLog_le_height N)
have hw := BinaryFraction.onlineParametersWithWork_width word hpositive
(Nat.size (inputSmallLog N)) hwidth
have hr := BinaryFraction.onlineParametersWithWork_range word hpositive (inputHeight N)
(by rw [hvalue]; exact computed_smallLog_le_height N)
have hc := online_parameter_width_cap _ _ hH hsmall
exact ⟨hw.1.trans hc.1, hw.2.trans hc.2, hr.1, hr.2⟩
theorem computedBinaryOnlineParameters_cost (N : ℕ) (word : List Bool)
(hwidth : word.length ≤ Nat.size (inputSmallLog N)) (hH : 64 ≤ inputHeight N)
(hpositive : 0 < bitWordValue word) :
(BinaryFraction.onlineParametersWithWork word hpositive).2 ≤ inputHeight N ^ 8 := by
have hsmall : Nat.size (inputSmallLog N) ≤ inputHeight N :=
(Nat.size_le.mpr Nat.lt_two_pow_self).trans (computed_smallLog_le_height N)
have hw : word.length ≤ inputHeight N := hwidth.trans hsmall
have hc := BinaryFraction.onlineParametersWithWork_cost word hpositive (inputHeight N) hw
have hbound (H : ℕ) (h : 64 ≤ H) :
bitPowerBitBudget 5 H + bitPowerBitBudget 4 H + 2 ≤ H ^ 8 := by
have h5 : 6 * (H + 1) + 1 ≤ 7 * H := by omega
have h4 : 5 * (H + 1) + 1 ≤ 6 * H := by omega
have hs5 := Nat.pow_le_pow_left h5 2
have hs4 := Nat.pow_le_pow_left h4 2
have hc : 5 * (200 * (6 * (H + 1) + 1) ^ 2) +
4 * (200 * (5 * (H + 1) + 1) ^ 2) + 4 ≤ 100000 * H ^ 2 := by nlinarith
have h6 : 100000 ≤ H ^ 6 := by
exact (by norm_num : 100000 ≤ 64 ^ 6).trans (Nat.pow_le_pow_left h 6)
change 1 + 5 * (200 * (6 * (H + 1) + 1) ^ 2) +
(1 + 4 * (200 * (5 * (H + 1) + 1) ^ 2)) + 2 ≤ H ^ 8
calc
_ ≤ 100000 * H ^ 2 := by omega
_ ≤ H ^ 6 * H ^ 2 := Nat.mul_le_mul_right _ h6
_ = _ := by ring
exact hc.trans (hbound _ hH)
end EditApproximation
end OAI
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