openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldConstruction /ConstructionFamily.lean
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.ConstructionData | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.Selection | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.RateLimit | |
| namespace OAI | |
| /-! Tensor extraction over arbitrary fields and its asymptotic rate. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.AllFieldConstructionFamily | |
| open MatrixMultiplication.Foundation AllFieldHistory AllFieldConstructionData | |
| open AllFieldConstructionRates Filter | |
| open scoped BigOperators Topology Classical | |
| attribute [local instance] Classical.propDecidable Classical.decEq | |
| variable {K : ℕ} | |
| theorem exists_width_eventually_data (allocation : Allocation) | |
| {slack widthMax : ℝ} (hslack : 0 < slack) (hmax : 0 < widthMax) : | |
| ∃ width : ℝ, 0 < width ∧ width ≤ widthMax ∧ | |
| ∀ᶠ m : ℕ in atTop, Nonempty (Data (K := K) allocation m width slack) := by | |
| let Index := Fin (K + 2) × Placement | |
| have hindex : (Finset.univ : Finset Index).Nonempty := | |
| ⟨(⟨0, by omega⟩, Equiv.refl _), Finset.mem_univ _⟩ | |
| have hb (i : Index) := | |
| AllFieldGroupSelection.exists_threshold_eventually_selected | |
| (K := K) (tick := i.1.val) allocation i.2 hslack hmax | |
| choose threshold hpositive hmaximum hselected using hb | |
| let width := Finset.univ.inf' hindex threshold | |
| have hwidth : 0 < width := by | |
| apply (Finset.lt_inf'_iff hindex).mpr | |
| intro i _ | |
| exact hpositive i | |
| have hbound (i : Index) : width ≤ threshold i := | |
| Finset.inf'_le _ (Finset.mem_univ i) | |
| have hwidthMax : width ≤ widthMax := | |
| (hbound hindex.choose).trans (hmaximum hindex.choose) | |
| have hgroups : ∀ᶠ m : ℕ in atTop, ∀ i : Index, | |
| Nonempty (AllFieldGroupSelection.Selected (K := K) (tick := i.1.val) | |
| allocation m width i.2 slack) := | |
| Filter.eventually_all.mpr (fun i => hselected i width hwidth (hbound i)) | |
| have hthresholds : ∀ᶠ m : ℕ in atTop, ∀ t : Fin (K + 2), | |
| AllFieldHistoryRecovery.minimumDilation (K := K) (tick := t.val) allocation width ≤ m := | |
| Filter.eventually_all.mpr (fun _ => eventually_ge_atTop _) | |
| refine ⟨width, hwidth, hwidthMax, ?_⟩ | |
| filter_upwards [AllFieldInitialChoice.eventually_nonempty (K := K) allocation hslack, | |
| hgroups, hthresholds] with m hinitial hgroup hthreshold | |
| exact ⟨{ initial := Classical.choice hinitial | |
| groups := fun t sigma => Classical.choice (hgroup (t, sigma)) | |
| threshold := hthreshold }⟩ | |
| def yieldRate (allocation : Allocation) (slack : ℝ) : ℝ := | |
| AllFieldInitialEntropy.nativeH0 + AllFieldScheduledYield.physicalYield K allocation / K - | |
| (1 + 6 * (K + 2) : ℕ) * slack / | |
| ((K : ℝ) * denominator (K := K) allocation) | |
| theorem exists_family (F : Type*) [Field F] (hK : 0 < K) | |
| (allocation : Allocation) {slack : ℝ} (hslack : 0 < slack) | |
| (hslots : ∀ slot : Slot K, slack ≤ (denominator (K := K) allocation : ℝ) * | |
| capacity allocation slot) : | |
| ∃ (W : ℕ → AllFieldWitness F) (scale : ℕ → ℝ), | |
| AllFieldRateLimit.HasRates W scale (yieldRate (K := K) allocation slack) | |
| AllFieldTerminalRates.S (8 * Real.log 7) := by | |
| obtain ⟨width, hwidth, _hmax, hd⟩ := exists_width_eventually_data | |
| (K := K) allocation hslack (show (0 : ℝ) < 1 by norm_num) | |
| have hterminal := (terminalDilation_tendsto K).eventually hd | |
| obtain ⟨M, hM⟩ := Filter.eventually_atTop.mp | |
| (hterminal.and (eventually_gt_atTop (0 : ℕ))) | |
| let data (n : ℕ) : Data (K := K) allocation | |
| (AllFieldTerminalRates.terminalDilation K (n + M)) width slack := | |
| Classical.choice (hM (n + M) (Nat.le_add_left M n)).1 | |
| have hn (n : ℕ) : 0 < n + M := (hM (n + M) (Nat.le_add_left M n)).2 | |
| let W (n : ℕ) := witness (F := F) allocation width slack (n + M) | |
| (hn n) hwidth hslots (data n) | |
| let scale (n : ℕ) : ℝ := (K : ℝ) * | |
| AllFieldTerminalRates.terminalLength allocation K (n + M) | |
| refine ⟨W, scale, ?_⟩ | |
| refine { scale_positive := ?_, yield_limit := ?_, volume_limit := ?_, rank_limit := ?_ } | |
| · apply Eventually.of_forall | |
| intro n | |
| exact mul_pos (Nat.cast_pos.mpr hK) | |
| (Nat.cast_pos.mpr (AllFieldTerminalRates.terminalLength_pos allocation (hn n))) | |
| · have h := (selected_terminal_rate hK allocation slack hslots).comp | |
| (tendsto_add_atTop_nat M) | |
| apply h.congr | |
| intro n | |
| simp only [Function.comp_def, W, scale, witness_multiplicity] | |
| · have h := (AllFieldTerminalRates.tendsto_terminalVolume hK allocation).comp | |
| (tendsto_add_atTop_nat M) | |
| apply h.congr | |
| intro n | |
| simp only [Function.comp_def, W, scale, witness_volume] | |
| · have h := (source_rank_terminal_rate hK allocation width).comp | |
| (tendsto_add_atTop_nat M) | |
| apply h.congr | |
| intro n | |
| simp only [Function.comp_def, W, scale, witness_rankBound] | |
| theorem fixed_prescription_constraint (F : Type*) [Field F] (hK : 0 < K) | |
| (allocation : Allocation) {slack : ℝ} (hslack : 0 < slack) | |
| (hslots : ∀ slot : Slot K, slack ≤ (denominator (K := K) allocation : ℝ) * | |
| capacity allocation slot) : | |
| yieldRate (K := K) allocation slack + (Arithmetic.omega F / 3) * | |
| AllFieldTerminalRates.S ≤ 8 * Real.log 7 := by | |
| obtain ⟨W, scale, h⟩ := exists_family F hK allocation hslack hslots | |
| exact AllFieldRateLimit.inequality_of_rates h | |
| theorem fixed_allocation_constraint (F : Type*) [Field F] (hK : 0 < K) | |
| (allocation : Allocation) (hcapacity : ∀ slot : Slot K, 0 < capacity allocation slot) : | |
| AllFieldInitialEntropy.nativeH0 + AllFieldScheduledYield.physicalYield K allocation / K + | |
| (Arithmetic.omega F / 3) * AllFieldTerminalRates.S ≤ 8 * Real.log 7 := by | |
| apply le_of_forall_pos_le_add | |
| intro η hη | |
| obtain ⟨slack, hslack, hslots, hloss⟩ := | |
| exists_slack_for_loss hK allocation hcapacity hη | |
| have h := fixed_prescription_constraint F hK allocation hslack hslots | |
| unfold yieldRate at h | |
| linarith | |
| end MatrixMultiplication.AllFieldConstructionFamily | |
| end | |
| end OAI | |