File size: 3,992 Bytes
cdec422 | 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 82 83 84 85 86 | import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.InitialFamily
import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.ConstructionLabels
namespace OAI
/-! Tensor extraction over arbitrary fields and its asymptotic rate. -/
noncomputable section
namespace MatrixMultiplication.AllFieldInitialChoice
open MatrixMultiplication.Foundation AllFieldHistory AllFieldFiniteFamily Filter
open scoped BigOperators Topology Classical
variable {K : ℕ}
structure Choice (allocation : Allocation) (m : ℕ) (δ : ℝ) where
sample : AllFieldInitialFamily.Sample (K := K) allocation
(δ / (AllFieldInitialFamily.grid (K := K) allocation : ℝ)) m
selected : Finset (JointPopulation.Target (initialJointCounts (K := K) allocation m))
card_selected : selected.card = AllFieldConstructionRates.initialCount (K := K) allocation δ m
good : ∀ e ∈ selected, AllFieldInitialFamily.Good allocation
(δ / (AllFieldInitialFamily.grid (K := K) allocation : ℝ)) m sample e
theorem initial_count_eq (allocation : Allocation) (m : ℕ) (δ : ℝ) :
⌊Real.exp ((populationLength (K := K) allocation m : ℝ) *
((K : ℝ) * AllFieldInitialEntropy.nativeH0 -
δ / (AllFieldInitialFamily.grid (K := K) allocation : ℝ)))⌋₊ =
AllFieldConstructionRates.initialCount (K := K) allocation δ m := by
rw [AllFieldConstructionRates.initialCount_eq,
AllFieldInitialFamily.populationLength_eq_mul_grid, Nat.cast_mul]
congr 2
change ((m : ℝ) * (AllFieldInitialFamily.grid (K := K) allocation : ℝ)) *
((K : ℝ) * AllFieldInitialEntropy.nativeH0 -
δ / (AllFieldInitialFamily.grid (K := K) allocation : ℝ)) =
(m : ℝ) * ((AllFieldInitialFamily.grid (K := K) allocation : ℝ) *
((K : ℝ) * AllFieldInitialEntropy.nativeH0) - δ)
have hD : (AllFieldInitialFamily.grid (K := K) allocation : ℝ) ≠ 0 :=
Nat.cast_ne_zero.mpr (AllFieldInitialFamily.grid_pos allocation).ne'
field_simp [hD]
theorem eventually_nonempty (allocation : Allocation) {δ : ℝ} (hδ : 0 < δ) :
∀ᶠ m : ℕ in atTop, Nonempty (Choice (K := K) allocation m δ) := by
have hs : 0 < δ / (AllFieldInitialFamily.grid (K := K) allocation : ℝ) :=
div_pos hδ (Nat.cast_pos.mpr (AllFieldInitialFamily.grid_pos allocation))
have h := AllFieldInitialFamily.eventually_exists_execution_with_width
(K := K) allocation ℚ 0 hs
filter_upwards [h] with m hm
obtain ⟨sample, selected, hcard, hgood, _execution⟩ := hm
exact ⟨{
sample := sample
selected := selected
card_selected := hcard.trans (initial_count_eq allocation m δ)
good := hgood }⟩
abbrev Tags (allocation : Allocation) (m : ℕ) (δ : ℝ)
(choice : Choice (K := K) allocation m δ) :=
AllFieldInitialHash.SelectedTargets (initialJointCounts (K := K) allocation m)
choice.selected
@[simp] theorem card_tags (allocation : Allocation) (m : ℕ) (δ : ℝ)
(choice : Choice (K := K) allocation m δ) :
Fintype.card (Tags allocation m δ choice) =
AllFieldConstructionRates.initialCount (K := K) allocation δ m := by
rw [AllFieldInitialHash.selectedTargets_card]
exact choice.card_selected
def execution (allocation : Allocation) (m : ℕ) (δ : ℝ)
(choice : Choice (K := K) allocation m δ) (F : Type*) [Field F] (ε : ℝ) :
Execution (cwSource F K (populationLength (K := K) allocation m))
(Tensor.directSum (fun _ : Tags allocation m δ choice =>
stateTensor F (K := K) allocation m ε 0)) :=
AllFieldInitialFamily.executionWithWidth allocation F
(δ / (AllFieldInitialFamily.grid (K := K) allocation : ℝ)) ε m
choice.sample choice.selected choice.good
@[simp] theorem execution_copies (allocation : Allocation) (m : ℕ) (δ : ℝ)
(choice : Choice (K := K) allocation m δ) (F : Type*) [Field F] (ε : ℝ) :
Fintype.card (execution allocation m δ choice F ε).Copies = 1 := rfl
end MatrixMultiplication.AllFieldInitialChoice
end
end OAI
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