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openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldGroups /TargetRate.lean
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roseking
Add lean source and binary files (part 84)
cdec422 verified
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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