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import OAI.LinearAlgebra.MatrixFields.Construction.GroupDegreeEnvelope
import OAI.LinearAlgebra.MatrixFields.Construction.GroupNativeDegreeBound
namespace OAI
namespace MatrixAllFields
open scoped BigOperators Topology Polynomial
section
noncomputable section
namespace MatrixMultiplication.AllFieldGroupSelection
open MatrixMultiplication.Foundation AllFieldHistory AllFieldActiveLaws
open AllFieldActiveCapacity AllFieldScheduledYield AllFieldNativeCapacity
open scoped BigOperators
attribute [local instance] Classical.propDecidable
variable {K tick : ℕ}
def orbitPrefactor (allocation : Allocation) (sigma : Placement) : ℕ :=
3 * (AllFieldHistoryOrbitCount.basePopulationBound (K := K) (tick := tick)
allocation + 1) ^ AllFieldGroupOrbitCount.groupDegree K tick sigma
theorem data_orbits_card_le (allocation : Allocation) (m : ℕ) (ε : ℝ)
(sigma : Placement)
(e : AllFieldGroupOrbitData.Targets (K := K) (tick := tick) allocation m sigma) :
((AllFieldGroupOrbitData.data (K := K) (tick := tick) allocation m ε sigma).orbits e).card ≤
orbitPrefactor (K := K) (tick := tick) allocation sigma *
(m + 1) ^ AllFieldGroupOrbitCount.groupDegree K tick sigma := by
have hb := AllFieldGroupOrbitData.orbits_card_le allocation m ε sigma e
unfold orbitPrefactor AllFieldGroupOrbitCount.groupOrbitPrefactor at *
exact_mod_cast hb
theorem data_coarse_injective (allocation : Allocation) (m : ℕ) (ε : ℝ)
(sigma : Placement) :
Function.Injective
(AllFieldGroupOrbitData.data (K := K) (tick := tick) allocation m ε sigma).coarse :=
AllFieldGroupOrbitData.coarse_injective allocation m sigma
structure Selected (allocation : Allocation) (m : ℕ) (width : ℝ)
(sigma : Placement) (slack : ℝ) where
sample : JointCoarseHashing.Sample
(AllFieldGroupOrbitData.Pos (K := K) (tick := tick) allocation m sigma)
(JointOrdinarySelection.hashLevel
(selectionHashRate (K := K) (tick := tick) allocation sigma slack) m)
targets : Finset (AllFieldGroupOrbitData.Targets
(K := K) (tick := tick) allocation m sigma)
card_targets : targets.card =
⌊Real.exp ((m : ℝ) * (nativeCapacity (K := K) (tick := tick) allocation - slack))⌋₊
good : ∀ e ∈ targets,
(AllFieldGroupOrbitData.data (K := K) (tick := tick) allocation m width sigma).Good
(JointOrdinarySelection.hashLevel
(selectionHashRate (K := K) (tick := tick) allocation sigma slack) m)
(JointOrdinarySelection.hashSet
(selectionHashRate (K := K) (tick := tick) allocation sigma slack) m)
(m : ℝ) e sample
distinct_first : ∀ e ∈ targets, ∀ f ∈ targets, e ≠ f →
(AllFieldGroupOrbitData.coarse (K := K) (tick := tick) allocation m sigma e).1 ≠
(AllFieldGroupOrbitData.coarse (K := K) (tick := tick) allocation m sigma f).1
end MatrixMultiplication.AllFieldGroupSelection
end
end
end MatrixAllFields
namespace MatrixAllFields
open scoped BigOperators Topology Polynomial
section
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) allocation sigma
(show 0 < slack / 4 by linarith) 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
end MatrixAllFields
end OAI
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