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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 | import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.TickCarry
import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.SourceAssignment
import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.Produced
import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.SelectionCore
import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.AssignmentRetention
import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.Assignments
namespace OAI
/-! Finite extraction histories, inherited masks and recovery bounds. -/
noncomputable section
namespace MatrixMultiplication.AllFieldHistoryTick
open MatrixMultiplication.Foundation AllFieldHistory AllFieldFiniteFamily
open AllFieldGroupSelection AllFieldHistoryGroupedRecovery
open scoped BigOperators Classical
attribute [local instance] Classical.propDecidable Classical.decEq
variable {K tick : ℕ} (allocation : Allocation) (m : ℕ) (ε slack : ℝ)
abbrev Choices := ∀ sigma : Placement,
Selected (K := K) (tick := tick) allocation m ε sigma slack
abbrev AmbientTag (sigma : Placement) :=
JointCoarseHashing.Triple
(AllFieldGroupOrbitData.Pos (K := K) (tick := tick) allocation m sigma)
abbrev Tags (chosen : Choices (K := K) (tick := tick) allocation m ε slack) :=
∀ sigma : Placement, {e // e ∈ (chosen sigma).targets}
local instance tagsDecidableEq (chosen : Choices (K := K) (tick := tick) allocation m ε slack) :
DecidableEq (Tags allocation m ε slack chosen) := Classical.decEq _
@[instance_reducible]
private def coarseTripleFintype (P : Type) [Fintype P] :
Fintype (JointCoarseHashing.Triple P) := inferInstance
instance ambientTagFintype (sigma : Placement) :
Fintype (AmbientTag (K := K) (tick := tick) allocation m sigma) :=
coarseTripleFintype (AllFieldGroupOrbitData.Pos (K := K) (tick := tick) allocation m sigma)
instance ambientTagsFintype :
Fintype (∀ sigma, AmbientTag (K := K) (tick := tick) allocation m sigma) := inferInstance
instance tagsFintype (chosen : Choices (K := K) (tick := tick) allocation m ε slack) :
Fintype (Tags allocation m ε slack chosen) := inferInstance
def select (chosen : Choices (K := K) (tick := tick) allocation m ε slack)
(j : Tags allocation m ε slack chosen) :
∀ sigma, AmbientTag (K := K) (tick := tick) allocation m sigma :=
fun sigma => AllFieldGroupOrbitData.coarse allocation m sigma (j sigma).val
theorem select_injective
(chosen : Choices (K := K) (tick := tick) allocation m ε slack) :
Function.Injective (select allocation m ε slack chosen) := by
intro _ _ hequal
funext sigma
apply Subtype.ext
exact AllFieldGroupOrbitData.coarse_injective allocation m sigma (congrFun hequal sigma)
def targets (chosen : Choices (K := K) (tick := tick) allocation m ε slack)
(j : Tags allocation m ε slack chosen) :
AllFieldHistoryRecovery.Targets (K := K) (tick := tick) allocation m :=
joinGroupTargets (activeCounts allocation m) (fun sigma => (j sigma).val)
def assignments (chosen : Choices (K := K) (tick := tick) allocation m ε slack)
(sigma : Placement) :=
AllFieldGroupAssignmentRetention.physicalAssignment allocation m ε sigma
(JointOrdinarySelection.hashLevel
(selectionHashRate (K := K) (tick := tick) allocation sigma slack) m)
(JointOrdinarySelection.hashSet
(selectionHashRate (K := K) (tick := tick) allocation sigma slack) m)
(chosen sigma).sample
def assignment (chosen : Choices (K := K) (tick := tick) allocation m ε slack) :=
groupedAssignment allocation m (AmbientTag (K := K) (tick := tick) allocation m)
(assignments allocation m ε slack chosen)
theorem coherent (F : Type*) [Field F]
(chosen : Choices (K := K) (tick := tick) allocation m ε slack) :
JointExtraction.Coherent
(AllFieldHistorySource.preparedSource (K := K) (tick := tick) allocation m F ε)
(assignment allocation m ε slack chosen) := by
apply AllFieldHistorySource.groupedAssignment_coherent allocation m F ε
intro sigma
exact AllFieldGroupAssignments.physicalAssignment_coherent F allocation m ε sigma
_ _ (chosen sigma).sample
theorem repair (F : Type*) [Field F] (hε : 0 < ε)
(hm : AllFieldHistoryRecovery.minimumDilation (K := K) (tick := tick) allocation ε ≤ m)
(chosen : Choices (K := K) (tick := tick) allocation m ε slack)
(j : Tags allocation m ε slack chosen) :
InverseLinearRecovery.RecoveredBy
(AllFieldHistoryRecovery.ideal F allocation m ε (targets allocation m ε slack chosen j))
(JointExtraction.branch
(AllFieldHistorySource.preparedSource (K := K) (tick := tick) allocation m F ε)
(assignment allocation m ε slack chosen) (select allocation m ε slack chosen j))
(AllFieldHistoryRecovery.shiftCount (K := K) (tick := tick) allocation ε m) := by
rw [AllFieldHistorySource.preparedSource_eq_delete]
apply repair_grouped F allocation hε hm
(AmbientTag (K := K) (tick := tick) allocation m)
(assignments allocation m ε slack chosen)
(select allocation m ε slack chosen j) (fun sigma => (j sigma).val)
· intro sigma w hw
have hs := AllFieldGroupAssignmentRetention.assigned_some_idealSide
allocation m ε sigma _ _ (chosen sigma).sample (j sigma).val (sigma.symm 0, w) hw
simpa only [Equiv.apply_symm_apply] using hs
· intro sigma w hw
have hs := AllFieldGroupAssignmentRetention.assigned_some_idealSide
allocation m ε sigma _ _ (chosen sigma).sample (j sigma).val (sigma.symm 1, w) hw
simpa only [Equiv.apply_symm_apply] using hs
· intro sigma w hw
have hs := AllFieldGroupAssignmentRetention.assigned_some_idealSide
allocation m ε sigma _ _ (chosen sigma).sample (j sigma).val (sigma.symm 2, w) hw
simpa only [Equiv.apply_symm_apply] using hs
· intro side w hw sigma
have hg := (chosen sigma).good (j sigma).val (j sigma).property
have hh := AllFieldGroupAssignmentRetention.good_projected_failure_le F
allocation m ε sigma _ _
(JointOrdinarySelection.hashSet_AP (selectionHashRate allocation sigma slack) m)
(m : ℝ) (chosen sigma).sample (targets allocation m ε slack chosen j)
(by simpa only [targets, project_joinGroupTargets] using hg) side w hw
simp only [targets, project_joinGroupTargets] at hh
rw [filter_eq_rejected] at hh ⊢
dsimp only [assignments, select]
exact hh
def canonicalMap (F : Type*) [Field F]
(e : AllFieldHistoryRecovery.Targets (K := K) (tick := tick) allocation m) :
LocalMap (AllFieldHistoryRecovery.ideal F allocation m ε e)
(AllFieldHistoryRecovery.canonical (K := K) (tick := tick) F allocation m ε) := by
apply LocalMap.ofExists
refine ⟨_, _, _, (Tensor.pullback_eq_restrict
(AllFieldHistoryRecovery.coordinates allocation m e).symm
(AllFieldHistoryRecovery.coordinates allocation m e).symm
(AllFieldHistoryRecovery.coordinates allocation m e).symm _).symm.trans ?_⟩
funext x y z
change AllFieldHistoryRecovery.ideal F allocation m ε e _ _ _ = _
rw [AllFieldHistoryRecovery.ideal_coordinates]
simp only [Equiv.apply_symm_apply]
def activeExecution (F : Type*) [Field F] (hε : 0 < ε)
(hm : AllFieldHistoryRecovery.minimumDilation (K := K) (tick := tick) allocation ε ≤ m)
(chosen : Choices (K := K) (tick := tick) allocation m ε slack) :
Execution
(AllFieldHistoryRegrouping.activeTensor (K := K) (tick := tick) allocation m F ε)
(Tensor.directSum (fun _ : Tags allocation m ε slack chosen =>
AllFieldHistoryRegrouping.producedTensor (K := K) (tick := tick) allocation m F ε)) := by
let prepared := (Execution.initial
(AllFieldHistoryRegrouping.activeTensor (K := K) (tick := tick) allocation m F ε)).restrict _
(AllFieldHistorySource.preparedMap (K := K) (tick := tick) allocation m F ε)
let repaired := prepared.extractSelectedAndRepair
(assignment allocation m ε slack chosen) (coherent allocation m ε slack F chosen)
(select allocation m ε slack chosen) (select_injective allocation m ε slack chosen)
(fun j => AllFieldHistoryRecovery.ideal F allocation m ε (targets allocation m ε slack chosen j))
(AllFieldHistoryRecovery.shiftCount (K := K) (tick := tick) allocation ε m)
(repair allocation m ε slack F hε hm chosen)
let canonical := repaired.restrict _ (LocalMap.directSum _ _
(fun j => canonicalMap allocation m ε F (targets allocation m ε slack chosen j)))
exact canonical.restrict _
((AllFieldHistoryProduced.map (K := K) (tick := tick) allocation m F hε.le).parallelCopies _)
@[simp] theorem activeExecution_copies (F : Type*) [Field F] (hε : 0 < ε)
(hm : AllFieldHistoryRecovery.minimumDilation (K := K) (tick := tick) allocation ε ≤ m)
(chosen : Choices (K := K) (tick := tick) allocation m ε slack) :
Fintype.card (activeExecution allocation m ε slack F hε hm chosen).Copies =
2 ^ (3 * AllFieldHistoryRecovery.shiftCount (K := K) (tick := tick) allocation ε m) := by
change Fintype.card (InverseLinearRecovery.MaskRectangles
(AllFieldHistoryRecovery.shiftCount (K := K) (tick := tick) allocation ε m) × PUnit) = _
rw [Fintype.card_prod, ExactRecovery.card_mask_rectangles, Fintype.card_fin,
Fintype.card_punit, mul_one]
theorem card_tags (chosen : Choices (K := K) (tick := tick) allocation m ε slack) :
Fintype.card (Tags allocation m ε slack chosen) =
∏ _sigma : Placement,
⌊Real.exp ((m : ℝ) * (nativeCapacity (K := K) (tick := tick) allocation - slack))⌋₊ := by
rw [Fintype.card_pi]
apply Finset.prod_congr rfl
intro sigma _
simpa only [Fintype.card_coe] using (chosen sigma).card_targets
def execution (F : Type*) [Field F] (hε : 0 < ε)
(hm : AllFieldHistoryRecovery.minimumDilation (K := K) (tick := tick) allocation ε ≤ m)
(chosen : Choices (K := K) (tick := tick) allocation m ε slack) :
Execution (stateTensor F (K := K) allocation m ε tick)
(Tensor.directSum (fun _ : Tags allocation m ε slack chosen =>
stateTensor F (K := K) allocation m ε (tick + 1))) :=
AllFieldHistoryTickCarry.liftExecution (K := K) (tick := tick) allocation m F ε
(activeExecution allocation m ε slack F hε hm chosen)
@[simp] theorem execution_copies (F : Type*) [Field F] (hε : 0 < ε)
(hm : AllFieldHistoryRecovery.minimumDilation (K := K) (tick := tick) allocation ε ≤ m)
(chosen : Choices (K := K) (tick := tick) allocation m ε slack) :
Fintype.card (execution allocation m ε slack F hε hm chosen).Copies =
2 ^ (3 * AllFieldHistoryRecovery.shiftCount (K := K) (tick := tick) allocation ε m) :=
activeExecution_copies allocation m ε slack F hε hm chosen
end MatrixMultiplication.AllFieldHistoryTick
end
end OAI
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