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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 | import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.CountRate
import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.PairEntropy
import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.ProjectedEntropy
namespace OAI
/-! Group assignments, orbit counts and extraction capacities. -/
noncomputable section
namespace MatrixMultiplication.AllFieldGroupNativeDegreeBound
open AllFieldHistory AllFieldActiveLaws AllFieldActiveCapacity
open AllFieldGroupOrbitData AllFieldGroupDegrees AllFieldGroupDegreeLaws
open AllFieldGroupCountRate AllFieldGroupPairWindows AllFieldGroupPairEntropy
open AllFieldGroupProjectedEntropy JointCompatibilityRateLimit
open MatrixMultiplication.Foundation Filter
open scoped BigOperators Topology
attribute [local instance] Classical.propDecidable Classical.decEq
variable {K tick : ℕ}
def maxNativeDegree (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 orderNativeDegree_le_max (allocation : Allocation) (sigma : Placement) (side : Fin 3) :
orderNativeDegree (K := K) (tick := tick) allocation sigma side ≤
maxNativeDegree (K := K) (tick := tick) allocation sigma := by
fin_cases side
· exact le_max_left _ _
· exact (le_max_left _ _).trans (le_max_right _ _)
· exact (le_max_right _ _).trans (le_max_right _ _)
theorem nativeCompatibilityRate_eq (allocation : Allocation) (sigma : Placement)
(side : Fin 3) (hside : side ≠ 0) :
nativeCompatibilityRate (K := K) (tick := tick) allocation sigma side =
orderNativeDegree (K := K) (tick := tick) allocation sigma side := by
have hp := projected_native_entropy_eq (K := K) (tick := tick) allocation sigma side hside
have hb := baseEntropy_sub_pairReferenceEntropy (K := K) (tick := tick) allocation sigma side
unfold nativeCompatibilityRate compatibilityRate
rw [show ∀ a b c d : ℝ, a - b + c + d = (a - b) + (c + d) by intros; ring,
hb, hp, ← Finset.sum_add_distrib]
unfold orderNativeDegree
apply Finset.sum_congr rfl
intro h _
have hs : (parentBase allocation h : ℝ) =
(populationLength (K := K) allocation 1 : ℝ) * orderMass allocation h :=
population_cast_real allocation 1 (h.val.val.1.source, h.val.val.2)
rw [hs]
ring
theorem exists_threshold_eventually_counts_le (allocation : Allocation) (sigma : Placement)
{δ widthMax : ℝ} (hδ : 0 < δ) (hwidthMax : 0 < widthMax) :
∃ threshold : ℝ, 0 < threshold ∧ threshold ≤ widthMax ∧
∀ width : ℝ, 0 < width → width ≤ threshold →
∀ᶠ m : ℕ in atTop,
(∀ e : Targets (K := K) (tick := tick) allocation m sigma,
(((data allocation m width sigma).eligibilityCompetitors e).card : ℝ) ≤
Real.exp ((m : ℝ) * (maxNativeDegree (K := K) (tick := tick) allocation sigma + δ))) ∧
(∀ (e : Targets (K := K) (tick := tick) allocation m sigma) (o : Orbit (K := K) (tick := tick) allocation m sigma),
o ∈ (data allocation m width sigma).orbits e →
∀ v : Variable (K := K) (tick := tick) allocation m sigma,
v ∈ (data allocation m width sigma).passing e o →
(((data allocation m width sigma).competitors e o v).card : ℝ) ≤
Real.exp ((m : ℝ) * (maxNativeDegree (K := K) (tick := tick) allocation sigma + δ))) := by
obtain ⟨t₁, ht₁, ht₁Max, h₁⟩ :=
exists_threshold_eventually_competitors_le (K := K) (tick := tick)
allocation sigma 1 hδ hwidthMax
obtain ⟨t₂, ht₂, _, h₂⟩ :=
exists_threshold_eventually_competitors_le (K := K) (tick := tick)
allocation sigma 2 hδ hwidthMax
refine ⟨min t₁ t₂, lt_min ht₁ ht₂, (min_le_left _ _).trans ht₁Max, ?_⟩
intro width hw hwt
filter_upwards [eventually_eligibility_card_le (K := K) (tick := tick)
allocation sigma width hδ,
h₁ width hw (hwt.trans (min_le_left _ _)),
h₂ width hw (hwt.trans (min_le_right _ _))] with m hm₀ hm₁ hm₂
have hm (side : Fin 3) :
Real.exp ((m : ℝ) * (orderNativeDegree (K := K) (tick := tick) allocation sigma side + δ)) ≤
Real.exp ((m : ℝ) * (maxNativeDegree (K := K) (tick := tick) allocation sigma + δ)) :=
Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left
(add_le_add_left (orderNativeDegree_le_max (K := K) (tick := tick) allocation sigma side) δ) (Nat.cast_nonneg m))
refine ⟨fun e => (hm₀ e).trans (hm 0), ?_⟩
intro e o ho v hv
rcases v with ⟨side, w⟩
fin_cases side
· exact (Nat.cast_le.mpr (Finset.card_le_card
(competitors_subset_eligibility_of_first allocation m width sigma e o (0, w) rfl))).trans
((hm₀ e).trans (hm 0))
· have hh := hm₁ e o ho w hv
rw [nativeCompatibilityRate_eq (K := K) (tick := tick) allocation sigma 1 (by decide)] at hh
exact hh.trans (hm 1)
· have hh := hm₂ e o ho w hv
rw [nativeCompatibilityRate_eq (K := K) (tick := tick) allocation sigma 2 (by decide)] at hh
exact hh.trans (hm 2)
end MatrixMultiplication.AllFieldGroupNativeDegreeBound
end
end OAI
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