File size: 5,363 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 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 | import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.SelectionCore
import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.MaskedGoodSelection
import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.DegreeEnvelope
import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.NativeDegreeBound
namespace OAI
/-! Group assignments, orbit counts and extraction capacities. -/
noncomputable section
namespace MatrixMultiplication.AllFieldGroupSelection
open MatrixMultiplication.Foundation AllFieldHistory AllFieldActiveLaws
open AllFieldActiveCapacity AllFieldScheduledYield AllFieldNativeCapacity
open Filter
open scoped BigOperators Topology
attribute [local instance] Classical.propDecidable
variable {K tick : ℕ}
private theorem selected_of_native_count_bound
(allocation : Allocation) (sigma : Placement) (width : ℝ)
{slack : ℝ} (hslack : 0 < slack)
(hcounts : ∀ᶠ m : ℕ in atTop,
(∀ e : AllFieldGroupOrbitData.Targets (K := K) (tick := tick) allocation m sigma,
(((AllFieldGroupOrbitData.data (K := K) (tick := tick) allocation m width sigma).eligibilityCompetitors e).card : ℝ) ≤
Real.exp ((m : ℝ) *
(nativeDegree (K := K) (tick := tick) allocation sigma + slack / 4))) ∧
(∀ e : AllFieldGroupOrbitData.Targets (K := K) (tick := tick) allocation m sigma,
∀ o ∈ (AllFieldGroupOrbitData.data (K := K) (tick := tick) allocation m width sigma).orbits e,
∀ v ∈ (AllFieldGroupOrbitData.data (K := K) (tick := tick) allocation m width sigma).passing e o,
(((AllFieldGroupOrbitData.data (K := K) (tick := tick) allocation m width sigma).competitors e o v).card : ℝ) ≤
Real.exp ((m : ℝ) *
(nativeDegree (K := K) (tick := tick) allocation sigma + slack / 4)))) :
∀ᶠ m : ℕ in atTop,
Nonempty (Selected (K := K) (tick := tick) allocation m width sigma slack) := by
have hD : 0 ≤ nativeDegree (K := K) (tick := tick) allocation sigma + slack / 4 := by
linarith [nativeDegree_nonneg (K := K) (tick := tick) allocation sigma]
have hdegree := AllFieldGroupDegreeEnvelope.eventually_log_uniformDegree_le
allocation width sigma hD hcounts
have hrate : ∀ η : ℝ, 0 < η → ∀ᶠ m : ℕ in atTop,
Real.log (AllFieldGroupDegreeEnvelope.uniformDegree
(K := K) (tick := tick) allocation m width sigma : ℝ) / (m : ℝ) ≤
nativeDegree (K := K) (tick := tick) allocation sigma + slack / 4 + η := by
intro η hη
filter_upwards [hdegree] with m hm
linarith
let : ∀ m : ℕ, Fintype (AllFieldGroupOrbitData.Pos
(K := K) (tick := tick) allocation m sigma) := fun m => inferInstance
let : ∀ m : ℕ, Fintype (AllFieldGroupOrbitData.Targets
(K := K) (tick := tick) allocation m sigma) := fun m => inferInstance
have hselected := JointMaskedGoodSelection.eventually_exists_good_selection
(fun m => AllFieldGroupOrbitData.data (K := K) (tick := tick) allocation m width sigma)
(fun m => data_coarse_injective (K := K) (tick := tick) allocation m width sigma)
(fun m => AllFieldGroupDegreeEnvelope.uniformDegree (K := K) (tick := tick) allocation m width sigma)
(orbitPrefactor (K := K) (tick := tick) allocation sigma)
(AllFieldGroupOrbitCount.groupDegree K tick sigma)
(fun m => AllFieldGroupDegreeEnvelope.eligibility_card_le (K := K) (tick := tick) allocation m width sigma)
(fun m => AllFieldGroupDegreeEnvelope.competitors_card_le (K := K) (tick := tick) allocation m width sigma)
(fun m => data_orbits_card_le (K := K) (tick := tick) allocation m width sigma)
(selectionHashRate_nonneg (K := K) (tick := tick) allocation sigma hslack)
(selectionHashRate_window_gap (K := K) (tick := tick) allocation sigma hslack) hrate
(Eventually.of_forall (AllFieldGroupTargetRate.target_card_pos
(K := K) (tick := tick) (sigma := sigma) allocation))
(AllFieldGroupTargetRate.tendsto_log_target_card_div
(K := K) (tick := tick) (sigma := sigma) allocation)
(half_pos hslack)
filter_upwards [hselected] with m hm
obtain ⟨s, G, hcard, hgood, hdistinct⟩ := hm
refine ⟨{
sample := s
targets := G
card_targets := ?_
good := hgood
distinct_first := hdistinct }⟩
simpa only [selection_exponent] using hcard
theorem exists_threshold_eventually_selected
(allocation : Allocation) (sigma : Placement)
{slack widthMax : ℝ} (hslack : 0 < slack) (hwidthMax : 0 < widthMax) :
∃ threshold : ℝ, 0 < threshold ∧ threshold ≤ widthMax ∧
∀ width : ℝ, 0 < width → width ≤ threshold →
∀ᶠ m : ℕ in atTop,
Nonempty (Selected (K := K) (tick := tick) allocation m width sigma slack) := by
obtain ⟨threshold, hthreshold, hthresholdMax, hcounts⟩ :=
AllFieldGroupNativeDegreeBound.exists_threshold_eventually_counts_le
(K := K) (tick := tick) (δ := slack / 4) allocation sigma
(div_pos hslack (by norm_num)) hwidthMax
refine ⟨threshold, hthreshold, hthresholdMax, ?_⟩
intro width hwidth hsmall
apply selected_of_native_count_bound (K := K) (tick := tick) allocation sigma width hslack
simpa only [nativeDegree, AllFieldGroupNativeDegreeBound.maxNativeDegree] using
hcounts width hwidth hsmall
end MatrixMultiplication.AllFieldGroupSelection
end
end OAI
|