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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 | import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.DegreeLaws
import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.PairWindows
import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.CompatibilityCountProducer
import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.PairMixtureControls
namespace OAI
/-! Group assignments, orbit counts and extraction capacities. -/
noncomputable section
namespace MatrixMultiplication.AllFieldGroupCountRate
open AllFieldHistory AllFieldHistorySupport AllFieldHistoryChildLaws
open AllFieldGroupOrbitData AllFieldGroupDegrees AllFieldGroupDegreeLaws
open AllFieldGroupNativeLaws AllFieldGroupPairWindows
open JointCompatibilityControls JointCompatibilityScaling JointCompatibilityRateLimit
open JointPopulationCompatibility JointTypeCounts MatrixMultiplication.Foundation Filter
open scoped BigOperators Topology
attribute [local instance] Classical.propDecidable Classical.decEq
variable {K tick : ℕ}
def nativeCompatibilityRate (allocation : Allocation) (sigma : Placement) (side : Fin 3) : ℝ :=
compatibilityRate (base (K := K) (tick := tick) allocation sigma side)
(q allocation sigma side)
(classDesignated false side sigma) (classDesignated true side sigma)
(groupCap (base allocation sigma side) (classDesignated false side sigma)
(nativeLaw allocation false side sigma) 0)
(groupCap (base allocation sigma side) (classDesignated true side sigma)
(nativeLaw allocation true side sigma) 0)
theorem base_positive_counts (allocation : Allocation) (sigma : Placement) (side : Fin 3)
(c : Class (K := K) (tick := tick) sigma) (u : JointPopulation.Shape)
(hu : 0 < base allocation sigma side c u) :
0 < Counts (K := K) (tick := tick) allocation 1 sigma c.1 u := by
unfold base classCounts at hu
split_ifs at hu with hs
· exact hu
· exact (Nat.lt_irrefl 0 hu).elim
theorem nativeLaw_normalized (allocation : Allocation) (right : Bool)
(sigma : Placement) (side : Fin 3)
(c : Class (K := K) (tick := tick) sigma) (u : JointPopulation.Shape)
(hu : 0 < base allocation sigma side c u) :
∑ a, nativeLaw allocation right side sigma c u a = 1 := by
have hp := base_positive_counts allocation sigma side c u hu
simp only [nativeLaw,
supportedHalfLaw_eq_of_counts_pos allocation 1 right c.1.val u hp (sigma side)]
exact halfLawAt_normalized allocation 1 right c.1.val u hp (sigma side)
theorem classCounts_dilation (allocation : Allocation) (m : ℕ)
(sigma : Placement) (side : Fin 3) :
classCounts (Counts (K := K) (tick := tick) allocation m sigma) (sigma side) =
fun c u => m * base allocation sigma side c u := by
funext c u
change (if JointPopulation.shapeSide (sigma side) u = c.2 then
activeCounts allocation m c.1.val u else 0) =
m * (if JointPopulation.shapeSide (sigma side) u = c.2 then
activeCounts allocation 1 c.1.val u else 0)
rw [activeCounts_dilation]
split_ifs <;> simp
theorem exists_widths_eventually_nativeCount_le (allocation : Allocation) (sigma : Placement)
(side : Fin 3) {δ χMax ηMax : ℝ}
(hδ : 0 < δ) (hχMax : 0 < χMax) (hηMax : 0 < ηMax) :
∃ χ η : ℝ, 0 < χ ∧ χ ≤ χMax ∧ 0 < η ∧ η ≤ ηMax ∧
∀ᶠ m : ℕ in atTop,
∀ (e : Targets (K := K) (tick := tick) allocation m sigma)
(w : Raw (K := K) (tick := tick) allocation m sigma),
(∀ h j, JointPopulation.shapeSide (sigma side) ((e h).val j) =
ownWord allocation m sigma w ⟨h, j⟩) →
PairWindow (base allocation sigma side) (q allocation sigma side)
(Positions (K := K) (tick := tick) allocation m sigma w) (pairCounts allocation m sigma w) η →
(nativeCount allocation m χ sigma side w : ℝ) ≤
Real.exp ((m : ℝ) *
(nativeCompatibilityRate (K := K) (tick := tick) allocation sigma side + δ)) := by
have hparts : ∀ c : Class (K := K) (tick := tick) sigma,
Function.Injective (fun a : Symbol (K := K) (tick := tick) sigma c × Symbol (K := K) (tick := tick) sigma c => (a.1, a.2)) :=
fun _ => fun _ _ h => h
have hleft : ∀ (c : Class (K := K) (tick := tick) sigma) (a : Symbol (K := K) (tick := tick) sigma c),
(baseSize (base allocation sigma side) c : ℝ) *
(∑ b : {b : Symbol (K := K) (tick := tick) sigma c × Symbol (K := K) (tick := tick) sigma c // b.1 = a}, q allocation sigma side c b.val) =
totalCenterCount (base allocation sigma side) (nativeLaw allocation false side sigma) c a := by
intro c a
exact JointPairMixtureControls.pairMixture_left_balance
(base allocation sigma side) (nativeLaw allocation false side sigma)
(nativeLaw allocation true side sigma)
(nativeLaw_normalized allocation true sigma side) c a
have hright : ∀ (c : Class (K := K) (tick := tick) sigma) (a : Symbol (K := K) (tick := tick) sigma c),
(baseSize (base allocation sigma side) c : ℝ) *
(∑ b : {b : Symbol (K := K) (tick := tick) sigma c × Symbol (K := K) (tick := tick) sigma c // b.2 = a}, q allocation sigma side c b.val) =
totalCenterCount (base allocation sigma side) (nativeLaw allocation true side sigma) c a := by
intro c a
exact JointPairMixtureControls.pairMixture_right_balance
(base allocation sigma side) (nativeLaw allocation false side sigma)
(nativeLaw allocation true side sigma)
(nativeLaw_normalized allocation false sigma side) c a
obtain ⟨χ, η, hχ, hχMax', hη, hηMax', hbound⟩ :=
JointCompatibilityCountProducer.exists_widths_eventually_count_le
(base allocation sigma side) (q allocation sigma side)
(fun _ p => p.1) (fun _ p => p.2)
(classDesignated false side sigma) (classDesignated true side sigma)
(nativeLaw allocation false side sigma) (nativeLaw allocation true side sigma)
hparts
(fun c u a => (nativeLaw_bounds allocation false side sigma c u a).1)
(fun c u a => (nativeLaw_bounds allocation false side sigma c u a).2)
(fun c u a => (nativeLaw_bounds allocation true side sigma c u a).1)
(fun c u a => (nativeLaw_bounds allocation true side sigma c u a).2)
hleft hright hδ hχMax hηMax
refine ⟨χ, η, hχ, hχMax', hη, hηMax', ?_⟩
filter_upwards [hbound] with m hm
intro e w hown hwindow
rw [← classCounts_dilation allocation m sigma side] at hm
let target : FixedSideTargets (Counts (K := K) (tick := tick) allocation m sigma) (sigma side)
(ownWord allocation m sigma w) := ⟨e, hown⟩
have hb := (hm (Positions (K := K) (tick := tick) allocation m sigma w) (pairCounts allocation m sigma w)
(candidateEquiv (Counts (K := K) (tick := tick) allocation m sigma) (sigma side)
(ownWord allocation m sigma w) target)
(statisticElement (Counts (K := K) (tick := tick) allocation m sigma) (Letter (K := K) (tick := tick) sigma) (Letter (K := K) (tick := tick) sigma)
(Symbol (K := K) (tick := tick) sigma) (Symbol (K := K) (tick := tick) sigma) (projectedStatistic sigma) (projectedStatistic sigma)
(ownWord allocation m sigma w) w) hwindow).1
rw [nativeCount, compatibleTargetCount_eq_kernel]
exact hb
theorem exists_threshold_eventually_competitors_le (allocation : Allocation)
(sigma : Placement) (side : Fin 3) {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)
(o : Orbit (K := K) (tick := tick) allocation m sigma), o ∈ (data allocation m width sigma).orbits e →
∀ w : Raw (K := K) (tick := tick) allocation m sigma,
(side, w) ∈ (data allocation m width sigma).passing e o →
((competitors allocation m width sigma e o (side, w)).card : ℝ) ≤
Real.exp ((m : ℝ) *
(nativeCompatibilityRate (K := K) (tick := tick) allocation sigma side + δ)) := by
obtain ⟨χ, η, hχ, hχMax, hη, _, hbound⟩ :=
exists_widths_eventually_nativeCount_le (K := K) (tick := tick)
allocation sigma side (ηMax := 1) hδ hwidthMax zero_lt_one
let threshold := min χ (η / windowFactor (K := K) (tick := tick) allocation sigma)
have hf := windowFactor_pos (K := K) (tick := tick) allocation sigma
have ht : 0 < threshold := lt_min hχ (div_pos hη hf)
refine ⟨threshold, ht, (min_le_left _ _).trans hχMax, ?_⟩
intro width hw hwt
have hwχ : width ≤ χ := hwt.trans (min_le_left _ _)
have hww : windowFactor (K := K) (tick := tick) allocation sigma * width ≤ η := by
have hh := (le_div_iff₀ hf).mp (hwt.trans (min_le_right _ _))
simpa only [mul_comm] using hh
filter_upwards [hbound, eventually_gt_atTop (0 : ℕ)] with m hm hmpos
intro e o ho w hpass
have ha := AllFieldGroupAdmissible.full_admissible allocation m width sigma
e o ho (side, w) ((data allocation m width sigma).passing_subset e o ho hpass)
have hp := passing_pairWindow allocation hmpos hw.le sigma e o ho (side, w) hpass
have hpair : PairWindow (base allocation sigma side) (q allocation sigma side)
(Positions (K := K) (tick := tick) allocation m sigma w) (pairCounts allocation m sigma w) η :=
fun c hc a => (hp c hc a).trans hww
have hc := hm e w (fun h j => (ha.1.1 h j).symm) hpair
exact (Nat.cast_le.mpr (competitors_card_le_nativeCount allocation m hw.le hwχ sigma
e o (side, w))).trans hc
end MatrixMultiplication.AllFieldGroupCountRate
end
end OAI
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