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import OAI.Computability.PerfectCompleteness.Algebra.AffineDecoderMeeting
import OAI.Computability.PerfectCompleteness.Algebra.RightDecoderAffineSuccess
import OAI.Computability.PerfectCompleteness.Algebra.TensorRowAdviceLemmas
import OAI.Computability.PerfectCompleteness.Algebra.TensorSliceDimension
import OAI.Computability.PerfectCompleteness.Decoding.HierarchicalLeftDecoder
import OAI.Computability.PerfectCompleteness.Reduction.FixedRows
namespace OAI
namespace PerfectCompleteness.HierarchicalDecoderMeeting
noncomputable section
open scoped Classical TensorProduct
open TreeSourceSpaces HierarchicalArrays HierarchicalTensorSlice
open UniqueGamesTheorem.Foundations.Games
open UniqueGamesTheorem.Appendix.RankLevelFilter (linearMapFintype)
attribute [local instance] linearMapFintype
attribute [local instance] RightDecoder.characterFintype RightDecoder.scalarFintype
private theorem affine_meeting_probability_lower
{source native : Type*} [AddCommGroup source] [Module F2 source]
[AddCommGroup native] [Module F2 native]
[FiniteDimensional F2 source] [FiniteDimensional F2 native]
(embedding : native →ₗ[F2] source) (knownRows : Submodule F2 source)
(columns : Submodule F2 (Module.Dual F2 (source ⧸ knownRows)))
(center : Module.Dual F2 source) {dimension : Nat}
(hcolumns : Module.finrank F2 columns ≤ dimension)
[nativeFintype : Fintype (Module.Dual F2 native)]
(right : FiniteDistribution (Module.Dual F2 native)) :
letI : Fintype (Module.Dual F2 source) := LeftDecoder.dualFintype (V := source)
∀ (left : FiniteDistribution (Module.Dual F2 source))
(nativeColumns : Submodule F2 (Module.Dual F2 native)) (offset : Module.Dual F2 native),
left = LeftDecoder.affineLaw knownRows columns center →
nativeColumns = AffineDecoderMeeting.nativeQ embedding knownRows columns →
offset = AffineDecoderMeeting.nativeCenter embedding center →
(1 / (2 : ℝ) ^ dimension) * right.probability
(fun decoded => decide (decoded - offset ∈ nativeColumns)) ≤
(left.product right).probability
(fun pair => decide (embedding.dualMap pair.1 = pair.2)) := by
intro left nativeColumns offset hleft hcolumns' hoffset
rw [hleft, hcolumns', hoffset]
cases Subsingleton.elim nativeFintype (LeftDecoder.dualFintype (V := native))
exact AffineDecoderMeeting.meeting_probability_lower embedding knownRows columns center hcolumns right
variable {branch rows : Nat → Nat} {n t : Nat}
(slots : RecursiveSpaces.Slots branch n → Fin t → MixedSupport.Slot)
(upper : Nodes branch n) (lowerLevel : Nat)
(background : HierarchicalMatrixTable.Background (rows := rows) slots upper)
local instance rowSpaceFintype : Fintype (NodeEmbedding.RowSpace slots upper) :=
Fintype.ofFinite _
local instance valueFintype : Fintype
(Block rows upper × HierarchicalMatrixTable.SideOutput (rows := rows) upper) :=
Fintype.ofFinite _
local instance upperDualFintype : Fintype (Module.Dual F2 (NodeEmbedding.RowSpace slots upper)) :=
LeftDecoder.dualFintype (V := NodeEmbedding.RowSpace slots upper)
variable {Ω : Type*} [Fintype Ω]
(original : FiniteDistribution Ω) (arrays : Ω → Arrays slots rows)
(lowerEvent : Ω → Bool) (κ : ℝ)
(σ : KeyStrategy.Strategy (TreeCanonical.locationCount branch n t))
{r : Nat} (ρ : ℝ) (A : ManyGoodRows.RowMap (Block rows upper) r)
(known : HiddenBucketBias.VisibleDirection (LinearMap.ker A) → NodeEmbedding.NodeH slots upper)
def useful (X : HierarchicalFrozenTables.QuotientMatrix slots upper lowerLevel background) : Prop :=
HierarchicalUsefulness.mark slots upper lowerLevel original arrays lowerEvent κ
⟨background, X⟩ = true
def leftLaw : FiniteDistribution (Module.Dual F2 (NodeEmbedding.RowSpace slots upper)) :=
HierarchicalLeftDecoder.adviceLaw slots upper lowerLevel background σ
(useful slots upper lowerLevel background original arrays lowerEvent κ) r ρ A
(A.comp (MatrixRowQuotient.projectMatrix
(HierarchicalFrozenTables.knownRows slots upper lowerLevel background)
(visibleMatrix slots upper A known)))
theorem leftLaw_eq_affine
(hgood : VisibleGood slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known) :
leftLaw slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known =
LeftDecoder.affineLaw
(HierarchicalFrozenTables.knownRows slots upper lowerLevel background)
(witness slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known hgood).columnSpace
(correction slots upper lowerLevel background
(witness slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known hgood).value) := by
have h := LeftDecoder.adviceLaw_of_good (TreeCanonical.numberedSlots slots)
(NodeEmbedding.RowSpace slots upper) (HierarchicalMatrixTable.other slots upper background) σ
(HierarchicalFrozenTables.knownRows slots upper lowerLevel background)
(HierarchicalFrozenTables.sectionMap slots upper lowerLevel background)
(HierarchicalFrozenTables.sectionMap_spec slots upper lowerLevel background)
(useful slots upper lowerLevel background original arrays lowerEvent κ) r ρ A
(A.comp (MatrixRowQuotient.projectMatrix
(HierarchicalFrozenTables.knownRows slots upper lowerLevel background)
(visibleMatrix slots upper A known))) hgood
dsimp only [leftLaw, HierarchicalLeftDecoder.adviceLaw, LeftDecoder.law, LeftDecoder.center,
HierarchicalTensorSlice.witness, TensorSelectedSlice.witness, HierarchicalTensorSlice.correction,
HierarchicalTensorSlice.table, HierarchicalUsefulness.table, HierarchicalFrozenTables.table,
useful] at h ⊢
exact h
theorem columnSpace_eq_nativeQ
(hgood : VisibleGood slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known) :
columnSpace slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known hgood =
AffineDecoderMeeting.nativeQ (NodeEmbedding.embed slots upper)
(HierarchicalFrozenTables.knownRows slots upper lowerLevel background)
(witness slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known hgood).columnSpace :=
HierarchicalTensorSlice.columnSpace_eq_pulled slots upper lowerLevel background
original arrays lowerEvent κ σ ρ A known hgood
theorem nativeCorrection_eq_nativeCenter
(hgood : VisibleGood slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known) :
nativeCorrection slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known hgood =
AffineDecoderMeeting.nativeCenter (NodeEmbedding.embed slots upper)
(correction slots upper lowerLevel background
(witness slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known hgood).value) := rfl
variable (lower : Nodes branch n) (a : Block rows lower)
(repeats : Nat → Nat) (cut : OwnInputReference.Cut upper lower)
(input : OwnInputReference.Input slots rows upper lower (LinearMap.ker A) a)
theorem meeting_probability_ge_coset
(hgood : VisibleGood slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known)
(threshold : ℝ) :
(1 / (2 : ℝ) ^ r) *
(RightDecoder.law slots rows upper lower (LinearMap.ker A) a repeats cut σ input threshold).probability
(fun decoded => decide (decoded -
nativeCorrection slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known hgood ∈
columnSpace slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known hgood)) ≤
((leftLaw slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known).product
(RightDecoder.law slots rows upper lower (LinearMap.ker A) a repeats cut σ input threshold)).probability
(fun pair => decide ((NodeEmbedding.embed slots upper).dualMap pair.1 = pair.2)) := by
convert affine_meeting_probability_lower
(source := NodeEmbedding.RowSpace slots upper)
(native := OwnInputReference.UpperSpace slots upper)
(NodeEmbedding.embed slots upper)
(nativeFintype := RightDecoder.scalarFintype slots upper)
(HierarchicalFrozenTables.knownRows slots upper lowerLevel background)
(witness slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known hgood).columnSpace
(correction slots upper lowerLevel background
(witness slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known hgood).value)
(witness slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known hgood).column_count
(RightDecoder.law slots rows upper lower (LinearMap.ker A) a repeats cut σ input threshold)
(leftLaw slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known)
(columnSpace slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known hgood)
(nativeCorrection slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known hgood)
(leftLaw_eq_affine slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known hgood)
(columnSpace_eq_nativeQ slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known hgood)
(nativeCorrection_eq_nativeCenter slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known hgood)
theorem meeting_probability_lower
(hgood : VisibleGood slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known)
(hne : ∃ T, J slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known T = true)
(ρpred h₀ : ℝ) (hρpred : 0 < ρpred) (hh₀ : 0 < h₀)
(hrep : RecursiveSamplerBias.RepetitionsBalanced repeats)
(hslice : (7 / 8 : ℝ) ^ repeats (Nodes.height upper) ≤ h₀ / 2)
(hlist : (7 / 8 : ℝ) ^ repeats (Nodes.height upper) ≤ (ρpred * h₀ / 4) ^ 2 / 2)
(hexcluded : 1 / (2 : ℝ) ^ Module.finrank F2 (LinearMap.ker A) < ρpred / 8)
(hsize : h₀ ≤ 1 / (2 : ℝ) ^ (r * rows (Nodes.height upper)))
(hmatch : ρpred ≤ RightDecoderWitness.conditionalMatch slots rows upper lower
(LinearMap.ker A) a repeats cut σ input
(columnSpace slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known hgood)
(target slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known hgood hne)
(nativeCorrection slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known hgood)
(visibleOffset slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known hgood)) :
(1 / (2 : ℝ) ^ r) *
((ρpred * h₀ / 4) ^ 2 / (2 * (2 : ℝ) ^ rows (Nodes.height upper))) ≤
((leftLaw slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known).product
(RightDecoder.law slots rows upper lower (LinearMap.ker A) a repeats cut σ input
(ρpred * h₀ / 4))).probability
(fun pair => decide ((NodeEmbedding.embed slots upper).dualMap pair.1 = pair.2)) := by
have hdimension := TensorSliceDimension.equation_finrank_le (LinearMap.ker A)
(columnSpace slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known hgood)
(HierarchicalTensorSlice.finrank_columnSpace_le slots upper lowerLevel background
original arrays lowerEvent κ σ ρ A known hgood)
have hsize' : h₀ ≤ 1 / (2 : ℝ) ^ Module.finrank F2
((columnSpace slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known hgood) →ₗ[F2]
LinearMap.ker A) :=
hsize.trans (one_div_pow_le_one_div_pow_of_le (by norm_num : (1 : ℝ) ≤ 2) hdimension)
have hright := RightDecoderAffineSuccess.coset_probability_lower slots rows upper lower
(LinearMap.ker A) a repeats cut σ input ρpred h₀ hρpred hh₀ hrep hslice hlist hexcluded
(columnSpace slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known hgood)
hsize'
(target slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known hgood hne)
(nativeCorrection slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known hgood)
(visibleOffset slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known hgood) hmatch
exact (mul_le_mul_of_nonneg_left hright
(div_nonneg zero_le_one (pow_nonneg (by norm_num : (0 : ℝ) ≤ 2) r))).trans
(meeting_probability_ge_coset slots upper lowerLevel background original arrays lowerEvent κ σ ρ A known
lower a repeats cut input hgood (ρpred * h₀ / 4))
end
end PerfectCompleteness.HierarchicalDecoderMeeting
namespace PerfectCompleteness.HierarchicalFixedDecoderMeeting
noncomputable section
open scoped Classical TensorProduct
open TreeSourceSpaces HierarchicalArrays HierarchicalTensorSlice
open UpperParameterScalars
open UniqueGamesTheorem.Foundations.Games
open UniqueGamesTheorem.Appendix.RankLevelFilter (linearMapFintype)
attribute [local instance] linearMapFintype
attribute [local instance] RightDecoder.characterFintype RightDecoder.scalarFintype
theorem kernel_finrank_ge {ℓ r : Nat} (A : ManyGoodRows.RowMap (Fin ℓ → F2) r) :
ℓ - r ≤ Module.finrank F2 (LinearMap.ker A) := by
have hrange : Module.finrank F2 (LinearMap.range A) ≤ r := by
simpa only [Module.finrank_fin_fun] using (Submodule.finrank_le (LinearMap.range A))
have hdim := LinearMap.finrank_range_add_finrank_ker A
simp only [Module.finrank_fin_fun] at hdim
change Module.finrank F2 (LinearMap.range A) + Module.finrank F2 (LinearMap.ker A) = ℓ at hdim
omega
variable {δ : ℚ} (plan : FixedRows.Plan δ)
theorem repetitionsBalanced : RecursiveSamplerBias.RepetitionsBalanced (FixedRows.repeats plan) :=
RecursiveSamplerBias.repetitionsBalanced_of_six_le (FixedRows.repeats plan)
(fun k => plan.repeats_min (FixedRows.rows plan (k + 1)))
theorem slice_error (height : Nat) :
(7 / 8 : ℝ) ^ FixedRows.repeats plan height ≤
h plan.order (FixedRows.rows plan height) / 2 :=
(plan.repeats_error (FixedRows.rows plan height)).trans (min_le_left _ _)
theorem list_error (height : Nat) :
(7 / 8 : ℝ) ^ FixedRows.repeats plan height ≤
tau (InitialParameters.useful δ) plan.density plan.order (FixedRows.rows plan height) ^ 2 / 2 :=
(plan.repeats_error (FixedRows.rows plan height)).trans (min_le_right _ _)
theorem kernel_excluded (height : Nat)
(A : ManyGoodRows.RowMap (Fin (FixedRows.rows plan height) → F2) plan.order) :
1 / (2 : ℝ) ^ Module.finrank F2 (LinearMap.ker A) <
rhoPred (InitialParameters.useful δ) plan.density / 8 := by
have hker := kernel_finrank_ge A
apply (one_div_pow_le_one_div_pow_of_le (by norm_num : (1 : ℝ) ≤ 2) hker).trans_lt
simpa only [FixedRows.rows, DescendingRows.treeRows] using
plan.rows_correction (plan.depth - height)
variable (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 rowSpaceFintype : Fintype (NodeEmbedding.RowSpace slots upper) :=
Fintype.ofFinite _
local instance valueFintype : Fintype
(Block (FixedRows.rows plan) upper ×
HierarchicalMatrixTable.SideOutput (rows := FixedRows.rows plan) upper) :=
Fintype.ofFinite _
local instance upperDualFintype : Fintype (Module.Dual F2 (NodeEmbedding.RowSpace slots upper)) :=
LeftDecoder.dualFintype (V := NodeEmbedding.RowSpace slots upper)
variable {Ω : Type*} [Fintype Ω]
(original : FiniteDistribution Ω) (arrays : Ω → Arrays slots (FixedRows.rows plan))
(lowerEvent : Ω → Bool)
(σ : KeyStrategy.Strategy (TreeCanonical.locationCount branch plan.depth t))
(A : ManyGoodRows.RowMap (Block (FixedRows.rows plan) upper) plan.order)
(known : HiddenBucketBias.VisibleDirection (LinearMap.ker A) → NodeEmbedding.NodeH slots upper)
(lower : Nodes branch plan.depth) (a : Block (FixedRows.rows plan) lower)
(cut : OwnInputReference.Cut upper lower)
(input : OwnInputReference.Input slots (FixedRows.rows plan) upper lower (LinearMap.ker A) a)
include hδ
theorem meeting_probability_lower
(hgood : VisibleGood slots upper lowerLevel background original arrays lowerEvent
(InitialParameters.useful δ) σ plan.density A known)
(hne : ∃ T, J slots upper lowerLevel background original arrays lowerEvent
(InitialParameters.useful δ) σ plan.density A known T = true)
(hmatch : rhoPred (InitialParameters.useful δ) plan.density ≤
RightDecoderWitness.conditionalMatch slots (FixedRows.rows plan) upper lower
(LinearMap.ker A) a (FixedRows.repeats plan) cut σ input
(columnSpace slots upper lowerLevel background original arrays lowerEvent
(InitialParameters.useful δ) σ plan.density A known hgood)
(target slots upper lowerLevel background original arrays lowerEvent
(InitialParameters.useful δ) σ plan.density A known hgood hne)
(nativeCorrection slots upper lowerLevel background original arrays lowerEvent
(InitialParameters.useful δ) σ plan.density A known hgood)
(visibleOffset slots upper lowerLevel background original arrays lowerEvent
(InitialParameters.useful δ) σ plan.density A known hgood)) :
(1 / (2 : ℝ) ^ plan.order) *
(tau (InitialParameters.useful δ) plan.density plan.order
(FixedRows.rows plan (Nodes.height upper)) ^ 2 /
(2 * (2 : ℝ) ^ FixedRows.rows plan (Nodes.height upper))) ≤
((HierarchicalDecoderMeeting.leftLaw slots upper lowerLevel background original arrays lowerEvent
(InitialParameters.useful δ) σ plan.density A known).product
(RightDecoder.law slots (FixedRows.rows plan) upper lower (LinearMap.ker A) a
(FixedRows.repeats plan) cut σ input
(tau (InitialParameters.useful δ) plan.density plan.order
(FixedRows.rows plan (Nodes.height upper))))).probability
(fun pair => decide ((NodeEmbedding.embed slots upper).dualMap pair.1 = pair.2)) := by
simpa only [UpperParameterScalars.tau] using
(HierarchicalDecoderMeeting.meeting_probability_lower slots upper lowerLevel background
original arrays lowerEvent (InitialParameters.useful δ) σ plan.density A known
lower a (FixedRows.repeats plan) cut input hgood hne
(rhoPred (InitialParameters.useful δ) plan.density)
(h plan.order (FixedRows.rows plan (Nodes.height upper)))
(rhoPred_pos (InitialParameters.useful_pos hδ) plan.density_pos)
(h_pos plan.order (FixedRows.rows plan (Nodes.height upper)))
(repetitionsBalanced plan) (slice_error plan (Nodes.height upper))
(list_error plan (Nodes.height upper)) (kernel_excluded plan (Nodes.height upper) A)
le_rfl hmatch)
end
end PerfectCompleteness.HierarchicalFixedDecoderMeeting
end OAI
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