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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 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 | import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.ActiveLaws
import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.CompatibilityScaling
namespace OAI
/-! Group assignments, orbit counts and extraction capacities. -/
noncomputable section
namespace MatrixMultiplication.AllFieldGroupTargetRate
open AllFieldHistory AllFieldActiveLaws MatrixMultiplication.Foundation Filter
open scoped BigOperators Topology
attribute [local instance] Classical.propDecidable Classical.decEq
variable {K tick : ℕ} {sigma : Placement}
theorem source_unit_pos (allocation : Allocation) :
0 < populationLength (K := K) allocation 1 :=
AllFieldPopulationCounts.blockLength_pos (amount (K := K) allocation) (by decide)
theorem source_length_dilation (allocation : Allocation) (m : ℕ) :
populationLength (K := K) allocation m =
m * populationLength (K := K) allocation 1 :=
populationLength_dilation allocation m
theorem orderCounts_dilation (allocation : Allocation) (m : ℕ)
(h : ActiveOrder K tick sigma) (u : JointPopulation.Shape) :
orderCounts allocation m h u = m * orderCounts allocation 1 h u := by
apply Nat.cast_injective (R := ℝ)
rw [Nat.cast_mul, orderCounts_source_cast, orderCounts_source_cast,
populationLength_dilation, Nat.cast_mul]
ring
theorem orderCounts_dilation_fun (allocation : Allocation) (m : ℕ) :
orderCounts (K := K) (tick := tick) (sigma := sigma) allocation m =
fun h u => m * orderCounts allocation 1 h u := by
funext h u
exact orderCounts_dilation allocation m h u
theorem orderCounts_sum (allocation : Allocation) (m : ℕ)
(h : ActiveOrder K tick sigma) :
(∑ u, orderCounts allocation m h u) =
population allocation m (h.val.val.1.source, h.val.val.2) :=
jointCounts_sum allocation m h.val.val
theorem baseSize_eq_population (allocation : Allocation) (m : ℕ)
(h : ActiveOrder K tick sigma) :
JointCompatibilityScaling.baseSize (orderCounts allocation m) h =
population allocation m (h.val.val.1.source, h.val.val.2) :=
orderCounts_sum allocation m h
theorem baseSize_dilation (allocation : Allocation) (m : ℕ)
(h : ActiveOrder K tick sigma) :
JointCompatibilityScaling.baseSize (orderCounts allocation m) h =
m * JointCompatibilityScaling.baseSize (orderCounts allocation 1) h := by
simp only [JointCompatibilityScaling.baseSize]
simp_rw [orderCounts_dilation allocation m]
exact (Finset.mul_sum Finset.univ _ m).symm
theorem baseSize_cast_real (allocation : Allocation)
(h : ActiveOrder K tick sigma) :
(JointCompatibilityScaling.baseSize (orderCounts allocation 1) h : ℝ) =
(populationLength (K := K) allocation 1 : ℝ) * orderMass allocation h := by
rw [baseSize_eq_population, population_cast_real]
rfl
theorem baseSize_pos (allocation : Allocation)
(h : ActiveOrder K tick sigma) :
0 < JointCompatibilityScaling.baseSize (orderCounts allocation 1) h := by
rw [baseSize_eq_population]
exact work_source_population_pos allocation (by decide) h.val.val
theorem base_empirical_eq (allocation : Allocation)
(h : ActiveOrder K tick sigma) (u : JointPopulation.Shape) :
(orderCounts allocation 1 h u : ℝ) /
JointCompatibilityScaling.baseSize (orderCounts allocation 1) h =
(orderLaw h).mass u := by
rw [baseSize_eq_population]
exact jointCounts_ratio allocation (by decide) h.val.val u
def groupEntropyRate (allocation : Allocation) : ℝ :=
∑ h : ActiveOrder K tick sigma,
((populationLength (K := K) allocation 1 : ℝ) * orderMass allocation h) *
finiteEntropy (orderLaw h).mass
theorem groupEntropyRate_nonneg (allocation : Allocation) :
0 ≤ groupEntropyRate (K := K) (tick := tick) (sigma := sigma) allocation := by
apply Finset.sum_nonneg
intro h _
exact mul_nonneg
(mul_nonneg (Nat.cast_nonneg _) (orderMass_nonneg allocation h))
(FiniteLaw.entropy_nonneg (orderLaw h))
theorem baseEntropy_eq_groupEntropyRate (allocation : Allocation) :
JointCompatibilityScaling.baseEntropy
(orderCounts (K := K) (tick := tick) (sigma := sigma) allocation 1) =
groupEntropyRate (K := K) (tick := tick) (sigma := sigma) allocation := by
simp only [JointCompatibilityScaling.baseEntropy, groupEntropyRate]
simp only [base_empirical_eq]
simp only [baseSize_cast_real]
theorem target_card_pos (allocation : Allocation) (m : ℕ) :
0 < Fintype.card (JointPopulation.Target
(orderCounts (K := K) (tick := tick) (sigma := sigma) allocation m)) :=
Fintype.card_pos_iff.mpr (JointPopulation.target_nonempty _)
theorem tendsto_log_target_card_div (allocation : Allocation) :
Tendsto (fun m : ℕ =>
Real.log (Fintype.card (JointPopulation.Target
(orderCounts (K := K) (tick := tick) (sigma := sigma) allocation m)) : ℝ) /
(m : ℝ)) atTop
(𝓝 (groupEntropyRate (K := K) (tick := tick) (sigma := sigma) allocation)) := by
apply JointPopulationRates.tendsto_log_target_card_div_of_coordinate_rates
(fun m => orderCounts allocation m)
(fun h => (populationLength (K := K) allocation 1 : ℝ) * orderMass allocation h)
(fun h => (orderLaw h).mass)
· intro h
exact mul_nonneg (Nat.cast_nonneg _) (orderMass_nonneg allocation h)
· intro h
exact (orderLaw h).total
· exact orderCounts_dilation_rate allocation
theorem tendsto_log_targetSet_card_div (allocation : Allocation) :
Tendsto (fun m : ℕ =>
Real.log ((JointPopulation.targetSet
(orderCounts (K := K) (tick := tick) (sigma := sigma) allocation m)).card : ℝ) /
(m : ℝ)) atTop
(𝓝 (groupEntropyRate (K := K) (tick := tick) (sigma := sigma) allocation)) := by
simpa only [JointPopulation.target_card, JointPopulation.targetSet_card] using
tendsto_log_target_card_div (K := K) (tick := tick) (sigma := sigma) allocation
end MatrixMultiplication.AllFieldGroupTargetRate
end
end OAI
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