openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldGroups /SelectionParameters.lean
| import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.TargetRate | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.ActiveCapacity | |
| namespace OAI | |
| /-! Group assignments, orbit counts and extraction capacities. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.AllFieldGroupSelection | |
| open MatrixMultiplication.Foundation AllFieldHistory AllFieldActiveLaws | |
| open AllFieldActiveCapacity AllFieldScheduledYield AllFieldNativeCapacity | |
| open scoped BigOperators | |
| attribute [local instance] Classical.propDecidable Classical.decEq | |
| variable {K tick : ℕ} | |
| def nativeDegree (allocation : Allocation) (sigma : Placement) : ℝ := | |
| max (orderNativeDegree (K := K) (tick := tick) allocation sigma 0) | |
| (max (orderNativeDegree (K := K) (tick := tick) allocation sigma 1) | |
| (orderNativeDegree (K := K) (tick := tick) allocation sigma 2)) | |
| theorem targetEntropy_eq (allocation : Allocation) (sigma : Placement) : | |
| AllFieldGroupTargetRate.groupEntropyRate (K := K) (tick := tick) | |
| (sigma := sigma) allocation = | |
| orderEntropyRate (K := K) (tick := tick) allocation sigma := rfl | |
| theorem firstDegree_nonneg (allocation : Allocation) (sigma : Placement) : | |
| 0 ≤ orderNativeDegree (K := K) (tick := tick) allocation sigma 0 := by | |
| apply Finset.sum_nonneg | |
| intro h _ | |
| apply mul_nonneg | |
| · exact mul_nonneg (Nat.cast_nonneg _) (orderMass_nonneg allocation h) | |
| · rw [order_first_capacity h] | |
| exact sub_nonneg.mpr | |
| (JointOrdinaryPopulationSelection.sideMass_entropy_le (orderLaw h) (sigma 0)) | |
| theorem nativeDegree_nonneg (allocation : Allocation) (sigma : Placement) : | |
| 0 ≤ nativeDegree (K := K) (tick := tick) allocation sigma := | |
| (firstDegree_nonneg allocation sigma).trans (le_max_left _ _) | |
| def nativeCapacity (allocation : Allocation) : ℝ := | |
| (populationLength (K := K) allocation 1 : ℝ) * | |
| ((1 / 6 : ℝ) * Staggering.minCapacity | |
| (FiniteSchedule.tickCapacity K nativeLA nativeLB | |
| (nativeLC (fun i => (allocation.mass i : ℝ))) tick)) | |
| theorem targetEntropy_sub_degree (allocation : Allocation) (sigma : Placement) : | |
| AllFieldGroupTargetRate.groupEntropyRate (K := K) (tick := tick) | |
| (sigma := sigma) allocation - nativeDegree (K := K) (tick := tick) allocation sigma = | |
| nativeCapacity (K := K) (tick := tick) allocation := | |
| order_entropy_sub_native_degree K tick allocation sigma | |
| def selectionHashRate (allocation : Allocation) (sigma : Placement) (slack : ℝ) : ℝ := | |
| nativeDegree (K := K) (tick := tick) allocation sigma + slack / 2 | |
| theorem selectionHashRate_nonneg (allocation : Allocation) (sigma : Placement) | |
| {slack : ℝ} (hslack : 0 < slack) : | |
| 0 ≤ selectionHashRate (K := K) (tick := tick) allocation sigma slack := by | |
| unfold selectionHashRate | |
| linarith [nativeDegree_nonneg (K := K) (tick := tick) allocation sigma] | |
| theorem selectionHashRate_gap (allocation : Allocation) (sigma : Placement) | |
| {slack : ℝ} (hslack : 0 < slack) : | |
| nativeDegree (K := K) (tick := tick) allocation sigma < | |
| selectionHashRate (K := K) (tick := tick) allocation sigma slack := by | |
| unfold selectionHashRate | |
| linarith | |
| theorem selectionHashRate_window_gap (allocation : Allocation) (sigma : Placement) | |
| {slack : ℝ} (hslack : 0 < slack) : | |
| nativeDegree (K := K) (tick := tick) allocation sigma + slack / 4 < | |
| selectionHashRate (K := K) (tick := tick) allocation sigma slack := by | |
| unfold selectionHashRate | |
| linarith | |
| theorem selection_exponent (allocation : Allocation) (sigma : Placement) (slack : ℝ) : | |
| AllFieldGroupTargetRate.groupEntropyRate (K := K) (tick := tick) | |
| (sigma := sigma) allocation - | |
| selectionHashRate (K := K) (tick := tick) allocation sigma slack - slack / 2 = | |
| nativeCapacity (K := K) (tick := tick) allocation - slack := by | |
| rw [selectionHashRate] | |
| linarith [targetEntropy_sub_degree (K := K) (tick := tick) allocation sigma] | |
| end MatrixMultiplication.AllFieldGroupSelection | |
| end | |
| end OAI | |