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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 | import OAI.Combinatorics.EditApproximation.Coarse.CoarseLevelMagnitude
namespace OAI
namespace EditApproximation
namespace BinaryFraction
theorem coarseBitParameters_grid (H N k B q L d : ℕ)
(hH : 100 ≤ H) (hd : d ≤ H)
(hN : Nat.size N ≤ 2 * H) (hk : Nat.size k ≤ 2 * H)
(hB : Nat.size B ≤ 2 * H) (hq : Nat.size q ≤ 2 * H) (hL : Nat.size L ≤ 2 * H) :
Nat.size (coarseCommonDenominator 1 q L B d) ≤ 10 * H ^ 2 ∧
Nat.size (coarseLevelMagnitude N k B d + 2) + 2 * (10 * H ^ 2) + 1 ≤ H ^ 3 := by
have hsum : Nat.size L + Nat.size B + Nat.size q ≤ 6 * H := by omega
have hcoeff := Nat.mul_le_mul hd hsum
have hden := coarseCommonDenominator_bits 1 q L B d
rw [Nat.size_one] at hden
have hbaseprod := natural_product_bit_length 2 B
have hbasesum := natural_sum_bit_length (2 * B) 2
rw [show Nat.size 2 = 2 by decide] at hbaseprod hbasesum
have hbase : Nat.size (2 * B + 2) ≤ 2 * H + 5 := by omega
have hsumNk := natural_sum_bit_length N k
have hsumone := natural_sum_bit_length (N + k) 1
rw [Nat.size_one] at hsumone
have hnk : Nat.size (N + k + 1) ≤ 4 * H + 3 := by omega
have hmag := coarseLevelMagnitude_bits N k B d
have hmag2 := natural_sum_bit_length (coarseLevelMagnitude N k B d) 2
rw [show Nat.size 2 = 2 by decide] at hmag2
have hprod := Nat.mul_le_mul (show d + 1 ≤ H + 1 by omega) hbase
have hsq : H ≤ H ^ 2 := Nat.le_pow (by decide)
have hcube : 100 * H ^ 2 ≤ H ^ 3 := by
have hh := Nat.mul_le_mul_right (H ^ 2) hH
nlinarith only [hh]
constructor
· nlinarith only [hden, hq, hcoeff, hH, hsq]
· nlinarith only [hmag, hmag2, hnk, hprod, hsq, hcube, hH]
end BinaryFraction
end EditApproximation
end OAI
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