| 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 |
|
|
