File size: 3,554 Bytes
5d5048e | 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 | import OAI.Combinatorics.LowSpaceSubsetSum.Streams.MeasuredLog2
import OAI.Combinatorics.LowSpaceSubsetSum.Storage.MixerOrderExecution
namespace OAI
/-! # Integer mixer parameter arithmetic after the measured counts
The count logarithms are computed by bounded division loops. Sorting and
all four prime-width formulas then use only stored integers.
-/
namespace LowSpaceSubsetSum.MixerParameterExecution
open scoped BigOperators
def low (n half count : ℕ) : ℕ × ℕ :=
let bits := MeasuredLog2.run n count
(max (rationalLowParameter n half) (bits.1 / 2), bits.2 + 12)
theorem low_value (n half count : ℕ) (hc : count ≤ 2 ^ n) :
(low n half count).1 = lowParameterFromCount n half count := by
simp only [low, MeasuredLog2.run_value n count hc, lowParameterFromCount]
theorem low_work (n half count : ℕ) : (low n half count).2 ≤ 3 * n + 13 := by
have h := MeasuredLog2.run_work n count
change _ + 12 ≤ _
omega
structure Parameters where
order : Vector (Fin 3) 3
low : Vector ℕ 3
widths : Vector ℕ 4
def run (n : ℕ) (counts : Vector ℕ 3) : Parameters × ℕ :=
let order := MixerOrderExecution.run counts
let lows := Vector.ofFn (fun i : Fin 3 => low n (mixerSize n / 2) counts[order.1[i.val].val])
let lowValues := lows.map Prod.fst
let widths : Vector ℕ 4 := #v[lowValues[2], lowValues[1] - lowValues[2],
lowValues[0] - lowValues[1], extraParameterFromLow n (mixerSize n) (fun i => lowValues[i.val])]
(⟨order.1, lowValues, widths⟩, order.2 + (∑ i : Fin 3, lows[i.val].2) + 30)
theorem order_value (n : ℕ) (counts : Vector ℕ 3) (i : Fin 3) :
(run n counts).1.order[i.val] = mixerOrderFromCounts (fun j => counts[j.val]) i :=
MixerOrderExecution.run_value counts i
theorem low_values (n : ℕ) (counts : Vector ℕ 3) (hc : ∀ i : Fin 3, counts[i.val] ≤ 2 ^ n)
(i : Fin 3) :
(run n counts).1.low[i.val] = lowParameterFromCount n (mixerSize n / 2)
counts[(mixerOrderFromCounts (fun j => counts[j.val]) i).val] := by
simp only [run, Vector.getElem_map, Vector.getElem_ofFn,
low_value n _ _ (hc _), MixerOrderExecution.run_value]
theorem low_values_of_counts (n : ℕ) (counts : Vector ℕ 3)
(hc : ∀ i : Fin 3, counts[i.val] ≤ 2 ^ n) (f : Fin 3 → ℕ)
(hf : ∀ i : Fin 3, counts[i.val] = f i) (i : Fin 3) :
(run n counts).1.low[i.val] = lowParameterFromCount n (mixerSize n / 2)
(f (mixerOrderFromCounts f i)) := by
rw [low_values n counts hc]
have he : (fun j : Fin 3 => counts[j.val]) = f := funext hf
change lowParameterFromCount n (mixerSize n / 2)
((fun j : Fin 3 => counts[j.val]) (mixerOrderFromCounts (fun j => counts[j.val]) i)) = _
rw [he]
theorem widths_value (n : ℕ) (counts : Vector ℕ 3) :
(run n counts).1.widths = #v[(run n counts).1.low[2],
(run n counts).1.low[1] - (run n counts).1.low[2],
(run n counts).1.low[0] - (run n counts).1.low[1],
extraParameterFromLow n (mixerSize n) (fun i => (run n counts).1.low[i.val])] := rfl
theorem work (n : ℕ) (counts : Vector ℕ 3) : (run n counts).2 ≤ 9 * n + 120 := by
have h := Finset.sum_le_sum (fun i (_ : i ∈ (Finset.univ : Finset (Fin 3))) =>
low_work n (mixerSize n / 2) counts[(MixerOrderExecution.run counts).1[i.val].val])
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, smul_eq_mul] at h
change 40 + (∑ i : Fin 3,
(low n (mixerSize n / 2) counts[(MixerOrderExecution.run counts).1[i.val].val]).2) + 30 ≤ _
omega
end LowSpaceSubsetSum.MixerParameterExecution
end OAI
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