File size: 5,351 Bytes
2f887db | 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 | import OAI.Computability.PerfectCompleteness.Construction.BucketFullAgreement
import OAI.Computability.PerfectCompleteness.Decoding.HierarchicalLowerInverseLemmas
import OAI.Computability.PerfectCompleteness.Foundations.MultiplicationFormInjectiveLemmas
namespace OAI
section
namespace PerfectCompleteness.FixedLowerUniformCollision
noncomputable section
open scoped BigOperators Classical
open TreeSourceSpaces HierarchicalArrays
open UniqueGamesTheorem.Foundations.Games
attribute [local instance] UniqueGamesTheorem.Appendix.RankLevelFilter.linearMapFintype
private theorem expectation_uniform_eq_expect {A : Type*} [Fintype A] [Nonempty A]
(f : A → ℝ) :
(FiniteDistribution.uniform A).expectation f = 𝔼 x : A, f x := by
rw [FiniteDistribution.expectation_uniform, Fintype.expect_eq_sum_div_card]
section Uniform
variable {Ω Y : Type*} [Fintype Y]
(V : Submodule F2 (Ω → F2)) [Fintype V] [FiniteDimensional F2 V]
(ℓ : Nat) [Nonempty (BucketSampler.Direction ℓ)]
def uniformLaw : FiniteDistribution
(BucketSampler.Direction ℓ × (BucketMatrixResampling.Matrix V ℓ × V)) :=
(FiniteDistribution.uniform (BucketSampler.Direction ℓ)).product
((FiniteDistribution.uniform (BucketMatrixResampling.Matrix V ℓ)).product
(FiniteDistribution.uniform V))
def uniformCollision (f : BucketMatrixResampling.Matrix V ℓ → Option Y)
(x : BucketSampler.Direction ℓ × (BucketMatrixResampling.Matrix V ℓ × V)) : Bool :=
TwoResponseCollision.collision f
(x.2.1, EvaluationMatrix.shift V x.2.1 x.1.val x.2.2)
theorem uniform_collision_eq (f : BucketMatrixResampling.Matrix V ℓ → Option Y) :
(uniformLaw V ℓ).probability (uniformCollision V ℓ f) =
PartialTableInverse.nonzeroAgreement f := by
unfold uniformLaw uniformCollision
rw [BucketQuotientAgreement.probability_collision_eq_expectation]
simp only [FiniteDistribution.expectation_product, expectation_uniform_eq_expect]
calc
_ = FiniteRangeInverse.agreement f := by
unfold FiniteRangeInverse.agreement
refine Finset.expect_congr (ι := BucketSampler.Direction ℓ) ?_ ?_
· ext a
simp only [Finset.mem_univ]
intro a _
rfl
_ = _ := FiniteRangeInverse.agreement_eq_nonzeroAgreement f
end Uniform
variable {δ : ℚ} (plan : FixedRows.Plan δ) (hδ : 0 < δ)
{branch : Nat → Nat} {t : Nat}
(slots : RecursiveSpaces.Slots branch plan.depth → Fin t → MixedSupport.Slot)
(upper : Nodes branch plan.depth) (lowerLevel : Nat)
(background : HierarchicalMatrixTable.Background (rows := FixedRows.rows plan) slots upper)
local instance lowerHFintype (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) :
Fintype (HierarchicalDecoderTables.LowerH slots upper lowerLevel d) := Fintype.ofFinite _
local instance upperDualFintype :
Fintype (HierarchicalAllDecoderTables.UpperAnswer slots upper) :=
HierarchicalAllDecoderTables.upperDualFintype slots upper
variable (hbranch : ∀ k < plan.depth, 0 < branch k)
(σ : KeyStrategy.Strategy (TreeCanonical.locationCount branch plan.depth t))
(useful : (bg : HierarchicalMatrixTable.Background
(rows := FixedRows.rows plan) slots upper) →
HierarchicalFrozenTables.QuotientMatrix slots upper lowerLevel bg → Prop)
include hbranch σ useful
local instance directionNonempty (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) :
Nonempty (BucketSampler.Direction
(FixedRows.rows plan (Nodes.height (HierarchicalLeftDecoder.LowerNode upper lowerLevel d)))) :=
⟨BucketUniform.coordinateDirection ⟨0,
lt_of_lt_of_le Nat.zero_lt_one
(plan.rows_pos (plan.depth -
Nodes.height (HierarchicalLeftDecoder.LowerNode upper lowerLevel d)))⟩⟩
theorem supported_collision_lt {adviceRows : Nat}
(density : ℝ) (hdensity : 0 < density)
(d : HierarchicalFrozenTables.LowerNodes upper lowerLevel)
(upperMatrix : HierarchicalMatrixTable.Matrix (rows := FixedRows.rows plan) slots upper)
(A : ManyGoodRows.RowMap (Block (FixedRows.rows plan) upper) adviceRows)
(table : HierarchicalAllDecoderTables.Input
(rows := FixedRows.rows plan) slots upper lowerLevel adviceRows →
HierarchicalAllDecoderTables.UpperAnswer slots upper)
(hsupport : (HierarchicalAllDecoderTables.upperTableLaw slots upper lowerLevel σ useful
adviceRows density).weight table ≠ 0) :
(uniformLaw (HierarchicalDecoderTables.LowerH slots upper lowerLevel d)
(FixedRows.rows plan (Nodes.height (HierarchicalLeftDecoder.LowerNode upper lowerLevel d)))).probability
(uniformCollision _ _
(HierarchicalLowerInverse.matrixTable slots upper lowerLevel background d upperMatrix A
(HierarchicalAllDecoderTables.decodeTable slots upper lowerLevel hbranch adviceRows d
(FixedRows.cutoff plan hδ) table))) <
UpperParameterScalars.gamma (InitialParameters.useful δ) plan.density
(InitialParameters.inverse δ) plan.order
(FixedRows.rows plan (Nodes.height upper)) ^ 2 / 8 := by
rw [uniform_collision_eq]
exact HierarchicalFixedLowerInverse.supported_upper_matrixTable_agreement_lt
plan hδ slots upper lowerLevel background hbranch σ useful density hdensity
d upperMatrix A table hsupport
end
end PerfectCompleteness.FixedLowerUniformCollision
end
end OAI
|