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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 | import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.InitialChoice
import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.Iteration
import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.FinishedWitness
import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.ConstructionRates
namespace OAI
/-! Tensor extraction over arbitrary fields and its asymptotic rate. -/
noncomputable section
namespace MatrixMultiplication.AllFieldConstructionData
open MatrixMultiplication.Foundation AllFieldHistory AllFieldFiniteFamily
open scoped BigOperators Classical
attribute [local instance] Classical.propDecidable Classical.decEq
variable {K : ℕ}
structure Data (allocation : Allocation) (dilation : ℕ) (width slack : ℝ) where
initial : AllFieldInitialChoice.Choice (K := K) allocation dilation slack
groups : AllFieldHistoryIteration.ScheduleChoices
(K := K) allocation dilation width slack
threshold : ∀ t : Fin (K + 2),
AllFieldHistoryRecovery.minimumDilation (K := K) (tick := t.val) allocation width ≤ dilation
variable (allocation : Allocation) (dilation : ℕ) (width slack : ℝ)
abbrev Tags (data : Data (K := K) allocation dilation width slack) :=
AllFieldHistoryIteration.Tags allocation dilation width slack
(AllFieldInitialChoice.Tags allocation dilation slack data.initial) data.groups
instance tagsFintype (data : Data (K := K) allocation dilation width slack) :
Fintype (Tags allocation dilation width slack data) := inferInstance
def execution (F : Type*) [Field F] (hwidth : 0 < width)
(data : Data (K := K) allocation dilation width slack) :
Execution (cwSource F K (populationLength (K := K) allocation dilation))
(Tensor.directSum (fun _ : Tags allocation dilation width slack data =>
stateTensor F (K := K) allocation dilation width (K + 2))) :=
AllFieldHistoryIteration.execution allocation dilation width slack F hwidth
data.threshold data.groups
(AllFieldInitialChoice.execution allocation dilation slack data.initial F width)
theorem card_tags (data : Data (K := K) allocation dilation width slack) :
Fintype.card (Tags allocation dilation width slack data) =
Fintype.card (AllFieldConstructionRates.SelectedLabels
(K := K) allocation slack dilation) := by
rw [AllFieldHistoryIteration.card_tags, AllFieldInitialChoice.card_tags,
AllFieldConstructionRates.card_selected]
congr 1
apply Finset.prod_congr rfl
intro t _
apply Finset.prod_congr rfl
intro sigma _
exact (AllFieldConstructionRates.groupCount_eq_native allocation slack dilation t sigma).symm
theorem execution_copies (F : Type*) [Field F] (hwidth : 0 < width)
(data : Data (K := K) allocation dilation width slack) :
Fintype.card (execution allocation dilation width slack F hwidth data).Copies =
Fintype.card (AllFieldConstructionRates.Copies
(K := K) allocation width dilation) := by
rw [execution, AllFieldHistoryIteration.execution_copies,
AllFieldInitialChoice.execution_copies, one_mul]
simp only [AllFieldConstructionRates.Copies, Fintype.card_pi,
InverseLinearRecovery.MaskRectangles, ExactRecovery.card_mask_rectangles, Fintype.card_fin]
theorem tags_pos
(hslack : ∀ slot : AllFieldConstructionRates.Slot K,
slack ≤ (AllFieldConstructionRates.denominator (K := K) allocation : ℝ) *
AllFieldConstructionRates.capacity allocation slot)
(data : Data (K := K) allocation dilation width slack) :
0 < Fintype.card (Tags allocation dilation width slack data) := by
rw [card_tags]
exact AllFieldConstructionRates.selected_pos allocation slack hslack dilation
variable {F : Type*} [Field F] (m : ℕ)
def witness (hm : 0 < m) (hwidth : 0 < width)
(hslack : ∀ slot : AllFieldConstructionRates.Slot K,
slack ≤ (AllFieldConstructionRates.denominator (K := K) allocation : ℝ) *
AllFieldConstructionRates.capacity allocation slot)
(data : Data (K := K) allocation
(AllFieldTerminalRates.terminalDilation K m) width slack) : AllFieldWitness F :=
AllFieldFinishedWitness.witness allocation m hm hwidth.le
(execution allocation (AllFieldTerminalRates.terminalDilation K m) width slack F hwidth data)
(tags_pos allocation (AllFieldTerminalRates.terminalDilation K m) width slack hslack data)
@[simp] theorem witness_volume (hm : 0 < m) (hwidth : 0 < width)
(hslack : ∀ slot : AllFieldConstructionRates.Slot K,
slack ≤ (AllFieldConstructionRates.denominator (K := K) allocation : ℝ) *
AllFieldConstructionRates.capacity allocation slot)
(data : Data (K := K) allocation
(AllFieldTerminalRates.terminalDilation K m) width slack) :
(witness (F := F) allocation width slack m hm hwidth hslack data).volume =
AllFieldTerminalRates.terminalVolume (K := K) allocation m :=
AllFieldFinishedWitness.witness_volume _ _ _ _ _ _
@[simp] theorem witness_multiplicity (hm : 0 < m) (hwidth : 0 < width)
(hslack : ∀ slot : AllFieldConstructionRates.Slot K,
slack ≤ (AllFieldConstructionRates.denominator (K := K) allocation : ℝ) *
AllFieldConstructionRates.capacity allocation slot)
(data : Data (K := K) allocation
(AllFieldTerminalRates.terminalDilation K m) width slack) :
(witness (F := F) allocation width slack m hm hwidth hslack data).multiplicity =
Fintype.card (AllFieldConstructionRates.SelectedLabels
(K := K) allocation slack (AllFieldTerminalRates.terminalDilation K m)) := by
change Fintype.card (Tags _ _ _ _ data) = _
exact card_tags _ _ _ _ data
@[simp] theorem witness_rankBound (hm : 0 < m) (hwidth : 0 < width)
(hslack : ∀ slot : AllFieldConstructionRates.Slot K,
slack ≤ (AllFieldConstructionRates.denominator (K := K) allocation : ℝ) *
AllFieldConstructionRates.capacity allocation slot)
(data : Data (K := K) allocation
(AllFieldTerminalRates.terminalDilation K m) width slack) :
(witness (F := F) allocation width slack m hm hwidth hslack data).rankBound =
AllFieldSource.rankBudget
(AllFieldConstructionRates.Copies (K := K) allocation width
(AllFieldTerminalRates.terminalDilation K m)) K
(AllFieldTerminalRates.terminalLength allocation K m) := by
rw [witness]
erw [AllFieldFinishedWitness.witness_rankBound]
unfold AllFieldSource.rankBudget
rw [execution_copies]
end MatrixMultiplication.AllFieldConstructionData
end
end OAI
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