File size: 3,957 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 | 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
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