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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 | import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.IncomingCore
import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.MaskLaws
import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.RawMasks
import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.StageCIncoming
import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.Tensors
namespace OAI
/-! Finite extraction histories, inherited masks and recovery bounds. -/
noncomputable section
namespace MatrixMultiplication.AllFieldHistoryIncomingMasks
open MatrixMultiplication.Foundation AllFieldHistory AllFieldParameters
open AllFieldHistoryChildLaws AllFieldHistorySupport AllFieldHistoryMasks
open AllFieldHistoryMaskLaws AllFieldHistoryRawMasks InheritedMasks
open scoped BigOperators
attribute [local instance] Classical.propDecidable Classical.decEq
variable {K tick : ℕ}
theorem empiricalLaw_congr_decidable {P A : Type*} [Fintype P]
(d₁ d₂ : DecidableEq A) (w : P → A) (a : A) :
@empiricalLaw P A _ d₁ w a = @empiricalLaw P A _ d₂ w a := by
cases Subsingleton.elim d₁ d₂
rfl
theorem typeWindow_mono_decidable {P A : Type*} [Fintype P]
(d₁ d₂ : DecidableEq A) (ν : A → ℝ) {η δ : ℝ} (hη : η ≤ δ)
(w : P → A) (hw : @typeWindow P A _ d₁ ν η w) :
@typeWindow P A _ d₂ ν δ w := by
cases Subsingleton.elim d₁ d₂
intro a
exact (hw a).trans hη
theorem weight_append {m n : ℕ} (l : Fin m → Fin 7) (r : Fin n → Fin 7) :
CWStrands.weight (Fin.append l r) = CWStrands.weight l + CWStrands.weight r := by
simp [CWStrands.weight, Fin.sum_univ_add]
theorem strand_append {F : Type*} [CommRing F] {m n : ℕ}
(x y z : (Fin m → Fin 7) × (Fin n → Fin 7)) :
CWStrands.strand (F := F) (Fin (m + n))
(Fin.append x.1 x.2) (Fin.append y.1 y.2) (Fin.append z.1 z.2) =
CWStrands.strand (Fin m) x.1 y.1 z.1 * CWStrands.strand (Fin n) x.2 y.2 z.2 := by
simp [CWStrands.strand, Fin.prod_univ_add]
theorem shapeTensor_append {F : Type*} [CommRing F] {m n : ℕ} (g : Shape)
(x y z : (Fin m → Fin 7) × (Fin n → Fin 7)) :
CWStrands.shapeTensor (F := F) (Fin (m + n)) g
(Fin.append x.1 x.2) (Fin.append y.1 y.2) (Fin.append z.1 z.2) =
CWStrands.shapeTensor (Fin m ⊕ Fin n) g
(Sum.elim x.1 x.2) (Sum.elim y.1 y.2) (Sum.elim z.1 z.2) := by
simp only [CWStrands.shapeTensor, weight_append, CWStrands.weight_sum_elim,
strand_append, CWStrands.strand_sum_elim, Tensor.product]
theorem shapeTensor_joinHalves {F : Type*} [CommRing F] (w : Work K) (g : Shape)
(x y z : (Fin w.halfLength → Fin 7) × (Fin w.halfLength → Fin 7)) :
CWStrands.shapeTensor (F := F) (Fin (currentLength w.source)) g
(joinHalves w x) (joinHalves w y) (joinHalves w z) =
CWStrands.shapeTensor (Fin w.halfLength ⊕ Fin w.halfLength) g
(Sum.elim x.1 x.2) (Sum.elim y.1 y.2) (Sum.elim z.1 z.2) := by
cases w <;> exact shapeTensor_append g x y z
def defaultStatistic : (w : Work K) → w.Statistic
| .stageA _ => (0, 0)
| .stageB _ => (0 : Fin 6)
| .stageC _ => PUnit.unit
def choiceAt (h : Active K tick) (a : Statistic h) : SymbolChoice K tick :=
Function.update (fun g => defaultStatistic g.val.1) h a
@[simp] theorem choiceAt_self (h : Active K tick) (a : Statistic h) :
choiceAt h a h = a := by simp [choiceAt]
theorem rawPairCount_eq_wordPopulation (allocation : Allocation) (m : ℕ)
(i : MaskIndex K tick) (w : ActiveRawPairs (K := K) (tick := tick) allocation m) :
rawPairCount allocation m i w =
(wordPopulation (fun j => (statistic i.1 (w i.1 j).1,
statistic i.1 (w i.1 j).2)) (i.2.1 i.1, i.2.2 i.1) : ℝ) := by
simp only [rawPairCount, wordPopulation, Fintype.card_subtype,
Finset.natCast_card_filter, Prod.mk.injEq]
@[simp] theorem fourStatistic_append (l r : Fin 2 → Fin 7) :
CWCompleteStatistics.fourStatistic (Fin.append l r) =
(CWCompleteStatistics.twoStatistic l, CWCompleteStatistics.twoStatistic r) := rfl
@[simp] theorem append_singletons (l r : Fin 1 → Fin 7) :
Fin.append l r = ![l 0, r 0] := by
funext i
fin_cases i <;> rfl
theorem rawPass_stageB_typeWindow (allocation : Allocation) (m : ℕ) (ε : ℝ)
(side : Fin 3) (w : ActiveRawPairs (K := K) (tick := tick) allocation m)
(hw : rawPass allocation m ε side w) (h : APositive K) (phi : Placement)
(ht : FiniteSchedule.lotTick (Work.stageB h).lot (Work.stageB h).stage = tick) :
typeWindow (fun ij : PairSlot => (halfLaw (aShape h.val) (phi.symm side) ij : ℝ))
(pairWidth ε (stageBActive h phi ht))
(fun j => CWCompleteStatistics.fourStatistic
(joinHalves (.stageB h) (w (stageBActive h phi ht) j))) := by
intro ij
have hp := hw (stageBActive h phi ht,
choiceAt (stageBActive h phi ht) ij.1, choiceAt (stageBActive h phi ht) ij.2)
rw [center_stageB] at hp
simp only [choiceAt_self (stageBActive h phi ht) ij.1,
choiceAt_self (stageBActive h phi ht) ij.2] at hp
have hcard : Fintype.card (JointPopulation.Positions (activeCounts allocation m)
(stageBActive h phi ht)) = population allocation m
((stageBActive h phi ht).val.1.source, (stageBActive h phi ht).val.2) := by
exact (Fintype.card_congr
(sourcePositionsEquiv allocation m (stageBActive h phi ht))).trans
(Fintype.card_fin _)
have hcount : (wordPopulation (fun j =>
(statistic (stageBActive h phi ht) (w (stageBActive h phi ht) j).1,
statistic (stageBActive h phi ht) (w (stageBActive h phi ht) j).2)) ij : ℝ) =
rawPairCount allocation m (stageBActive h phi ht,
choiceAt (stageBActive h phi ht) ij.1, choiceAt (stageBActive h phi ht) ij.2) w := by
rcases ij with ⟨a, b⟩
simp only [wordPopulation, Fintype.card_subtype,
Finset.natCast_card_filter, rawPairCount]
apply Finset.sum_congr rfl
intro j _
apply if_congr
· simp only [choiceAt_self (stageBActive h phi ht) a,
choiceAt_self (stageBActive h phi ht) b]
exact Prod.mk_inj
· rfl
· rfl
have hword : (fun j => CWCompleteStatistics.fourStatistic
(joinHalves (.stageB h) (w (stageBActive h phi ht) j))) =
(fun j => (statistic (stageBActive h phi ht) (w (stageBActive h phi ht) j).1,
statistic (stageBActive h phi ht) (w (stageBActive h phi ht) j).2)) := by
funext j
rfl
have he : empiricalLaw (fun j => CWCompleteStatistics.fourStatistic
(joinHalves (.stageB h) (w (stageBActive h phi ht) j))) ij =
empiricalLaw (fun j =>
(statistic (stageBActive h phi ht) (w (stageBActive h phi ht) j).1,
statistic (stageBActive h phi ht) (w (stageBActive h phi ht) j).2)) ij := by
rw [hword]
exact empiricalLaw_congr_decidable _ _ _ _
rw [he]
unfold empiricalLaw
rw [hcard, hcount]
exact hp
attribute [local irreducible] activeCounts population
theorem incoming_stageA (allocation : Allocation) (m : ℕ) (ε : ℝ)
(side : Fin 3) (w : ActiveRawPairs (K := K) (tick := tick) allocation m)
(h : InitialPositive K) (phi : Placement)
(ht : FiniteSchedule.lotTick (Work.stageA h).lot (Work.stageA h).stage = tick) :
residentMask allocation m ε ((Work.stageA h).source, phi) side
(incomingWord allocation m ⟨(.stageA h, phi), ht⟩ w) := by
trivial
theorem incoming_stageB (allocation : Allocation) (m : ℕ) {ε : ℝ} (hε : 0 ≤ ε)
(side : Fin 3) (w : ActiveRawPairs (K := K) (tick := tick) allocation m)
(hw : rawPass allocation m ε side w) (h : APositive K) (phi : Placement)
(ht : FiniteSchedule.lotTick (Work.stageB h).lot (Work.stageB h).stage = tick) :
residentMask allocation m ε ((Work.stageB h).source, phi) side
(incomingWord allocation m (stageBActive h phi ht) w) := by
intro _
have hraw := rawPass_stageB_typeWindow allocation m ε side w hw h phi ht
have htransport := (JointCanonicalization.typeWindow_reindex
(fun ij : PairSlot => (halfLaw (aShape h.val) (phi.symm side) ij : ℝ))
(pairWidth ε (stageBActive h phi ht))
(fun j => CWCompleteStatistics.fourStatistic
(joinHalves (.stageB h) (w (stageBActive h phi ht) j)))
(sourcePositionsEquiv allocation m (stageBActive h phi ht)).symm).2 hraw
have hwidth : pairWidth ε (stageBActive h phi ht) ≤ ε / 8 := by
change ε / 16 ^ 1 ≤ ε / 8
norm_num
linarith
dsimp only [Work.source, residentLaw, residentLawRat, residentWidth, residentStatistic]
exact typeWindow_mono_decidable _ _ _ hwidth _ htransport
theorem incoming_stageC (allocation : Allocation) {m : ℕ} (hm : 0 < m)
{ε : ℝ} (hε : 0 ≤ ε) (side : Fin 3)
(e : JointPopulation.Target (activeCounts (K := K) (tick := tick) allocation m))
(w : ActiveRawPairs (K := K) (tick := tick) allocation m)
(h : PartC K) (phi : Placement)
(ht : FiniteSchedule.lotTick (Work.stageC h).lot (Work.stageC h).stage = tick)
(hweights : ∀ j,
CWLeafStatistics.weight ((w (AllFieldHistoryStageCIncoming.stageCActive h phi ht) j).1 (0 : Fin 1)) =
(JointPopulation.shapeSide side ((e (AllFieldHistoryStageCIncoming.stageCActive h phi ht)).val j)).val ∧
CWLeafStatistics.weight ((w (AllFieldHistoryStageCIncoming.stageCActive h phi ht) j).2 (0 : Fin 1)) =
activeParentShape (AllFieldHistoryStageCIncoming.stageCActive h phi ht) side -
(JointPopulation.shapeSide side ((e (AllFieldHistoryStageCIncoming.stageCActive h phi ht)).val j)).val) :
residentMask allocation m ε ((Work.stageC h).source, phi) side
(incomingWord allocation m (AllFieldHistoryStageCIncoming.stageCActive h phi ht) w) := by
intro _
have hraw := AllFieldHistoryStageCIncoming.typeWindow_of_target_side_weights
allocation hm h phi ht e w side hweights (ε / 256) (div_nonneg hε (by norm_num))
have hword :
(fun j => CWCompleteStatistics.twoStatistic
(joinHalves (.stageC h)
(w (AllFieldHistoryStageCIncoming.stageCActive h phi ht) j))) =
(fun j => CWCompleteStatistics.twoStatistic
![(w (AllFieldHistoryStageCIncoming.stageCActive h phi ht) j).1 (0 : Fin 1),
(w (AllFieldHistoryStageCIncoming.stageCActive h phi ht) j).2 (0 : Fin 1)]) := by
funext j
exact congrArg CWCompleteStatistics.twoStatistic (append_singletons _ _)
rw [← hword] at hraw
have htransport := (JointCanonicalization.typeWindow_reindex
(fun a : Fin 6 => (littleLaw (cParameterParent h) (cShapeParent h) (phi.symm side) a : ℝ))
(ε / 256)
(fun j => CWCompleteStatistics.twoStatistic
(joinHalves (.stageC h)
(w (AllFieldHistoryStageCIncoming.stageCActive h phi ht) j)))
(sourcePositionsEquiv allocation m (AllFieldHistoryStageCIncoming.stageCActive h phi ht)).symm).2 hraw
exact typeWindow_mono_decidable _ _ _ le_rfl _ htransport
def stageCWeights (allocation : Allocation) (m : ℕ) (side : Fin 3)
(e : JointPopulation.Target (activeCounts (K := K) (tick := tick) allocation m))
(w : ActiveRawPairs (K := K) (tick := tick) allocation m) : Prop :=
∀ (h : PartC K) (phi : Placement)
(ht : FiniteSchedule.lotTick (Work.stageC h).lot (Work.stageC h).stage = tick) j,
CWLeafStatistics.weight ((w (AllFieldHistoryStageCIncoming.stageCActive h phi ht) j).1 (0 : Fin 1)) =
(JointPopulation.shapeSide side ((e (AllFieldHistoryStageCIncoming.stageCActive h phi ht)).val j)).val ∧
CWLeafStatistics.weight ((w (AllFieldHistoryStageCIncoming.stageCActive h phi ht) j).2 (0 : Fin 1)) =
activeParentShape (AllFieldHistoryStageCIncoming.stageCActive h phi ht) side -
(JointPopulation.shapeSide side ((e (AllFieldHistoryStageCIncoming.stageCActive h phi ht)).val j)).val
theorem received_of_rawPass (allocation : Allocation) {m : ℕ} (hm : 0 < m)
{ε : ℝ} (hε : 0 ≤ ε) (side : Fin 3)
(e : JointPopulation.Target (activeCounts (K := K) (tick := tick) allocation m))
(w : ActiveRawPairs (K := K) (tick := tick) allocation m)
(hw : rawPass allocation m ε side w) (hc : stageCWeights allocation m side e w) :
received allocation m ε side w := by
rintro ⟨⟨work, phi⟩, ht⟩
cases work with
| stageA h => exact incoming_stageA allocation m ε side w h phi ht
| stageB h => exact incoming_stageB allocation m hε side w hw h phi ht
| stageC h => exact incoming_stageC allocation hm hε side e w h phi ht (hc h phi ht)
end MatrixMultiplication.AllFieldHistoryIncomingMasks
end
end OAI
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