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import OAI.Computability.PerfectCompleteness.Decoding.SourceQuestionCollisionTransportLemmas
import OAI.Computability.PerfectCompleteness.Decoding.SourceQuestionPhysicalCollision
namespace OAI
section
namespace PerfectCompleteness.FixedSourcePhysicalHigh
noncomputable section
open scoped Classical BigOperators
open FixedParameters FixedRows RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays
open UniqueGamesTheorem.Foundations.Games
variable {δ : ℚ} {hδ : 0 < δ} (parameters : Parameters δ hδ)
{height v m : Nat} [NeZero m]
(outside : Slots (branch parameters) parameters.plan.depth →
Fin (sourceLength parameters.plan hδ) → MixedSupport.Slot)
(placeholder : Slots (branch parameters) (height + 1) →
Fin (sourceLength parameters.plan hδ) → MixedSupport.Slot)
(clauses : Fin m → SourceClause.NormalizedClause v)
(designated : Fin (branch parameters height) → Slots (branch parameters) height)
(upper : Nodes (branch parameters) parameters.plan.depth)
(pDown : Path (branch parameters) (Nodes.height upper) (height + 1))
(hproper : height + 1 < Nodes.height upper)
(d : HierarchicalFrozenTables.LowerNodes upper (height + 1))
(hnode : WholeArrayInteriorExterior.upperNode ((Nodes.path upper).append pDown) =
HierarchicalLeftDecoder.LowerNode upper (height + 1) d)
{adviceRows : Nat}
(A : ManyGoodRows.RowMap (Block (rows parameters.plan) upper) adviceRows)
(cut : OwnInputReference.Cut upper (ProjectedLowerFiber.lower upper (height + 1) d))
(exterior : CleanPhysicalReplay.Exterior (rows parameters.plan) (repeats parameters.plan)
((Nodes.path upper).append pDown) outside placeholder)
(σ : KeyStrategy.Strategy (TreeCanonical.locationCount (branch parameters) parameters.plan.depth
(sourceLength parameters.plan hδ)))
(useful : (q : FixedSourceGlobalCollision.Questions (m := m) parameters (height := height)) →
(bg : HierarchicalMatrixTable.Background (rows := rows parameters.plan)
(SourceQuestionLowerForms.slots ((Nodes.path upper).append pDown) outside clauses q) upper) →
HierarchicalFrozenTables.QuotientMatrix
(SourceQuestionLowerForms.slots ((Nodes.path upper).append pDown) outside clauses q)
upper (height + 1) bg → Prop)
(density threshold : ℝ)
local instance rowSpaceFintype (q : FixedSourceGlobalCollision.Questions
(m := m) parameters (height := height)) :
Fintype (NodeEmbedding.RowSpace
(SourceQuestionLowerForms.slots ((Nodes.path upper).append pDown) outside clauses q) upper) :=
Fintype.ofFinite _
local instance upperAnswerFintype (q : FixedSourceGlobalCollision.Questions
(m := m) parameters (height := height)) :
Fintype (HierarchicalAllDecoderTables.UpperAnswer
(SourceQuestionLowerForms.slots ((Nodes.path upper).append pDown) outside clauses q) upper) :=
LeftDecoder.dualFintype
(V := NodeEmbedding.RowSpace
(SourceQuestionLowerForms.slots ((Nodes.path upper).append pDown) outside clauses q) upper)
local instance directionNonempty :
Nonempty (BucketSampler.Direction (rows parameters.plan (height + 1))) :=
⟨BucketUniform.coordinateDirection ⟨0, lt_of_lt_of_le Nat.zero_lt_one
(parameters.plan.rows_pos (parameters.plan.depth - (height + 1)))⟩⟩
def tagLaw : FiniteDistribution
(SourceProjectedTag.Tag (branch := branch parameters) (n := height)
(t := sourceLength parameters.plan hδ)) :=
SourceProjectedTag.law (FixedSourceGlobalCollision.flag parameters)
def success (direction : BucketSampler.Direction (rows parameters.plan (height + 1)))
(q : FixedSourceGlobalCollision.Questions (m := m) parameters (height := height))
(tag : SourceProjectedTag.Tag (branch := branch parameters) (n := height)
(t := sourceLength parameters.plan hδ)) : ℝ :=
SourceQuestionPhysicalHigh.success outside placeholder clauses designated q tag.1 tag.2 upper pDown
hproper (height + 1) (FixedLowerRawCollision.branchPositive parameters) d hnode A cut σ (useful q)
density threshold exterior direction (cutoff parameters.plan hδ)
def probability : ℝ :=
(FiniteDistribution.uniform (BucketSampler.Direction (rows parameters.plan (height + 1)))).expectation
(fun direction => (PreliminarySampler.questionsLaw (branch := branch parameters) (n := height + 1)
(t := sourceLength parameters.plan hδ) (m := m)).expectation (fun q =>
(tagLaw parameters).expectation
(success parameters outside placeholder clauses designated upper pDown hproper d hnode A cut
exterior σ useful density threshold direction q)))
theorem probability_square_le_collision :
probability parameters outside placeholder clauses designated upper pDown hproper d hnode A cut
exterior σ useful density threshold ^ 2 ≤
(FiniteDistribution.uniform (BucketSampler.Direction (rows parameters.plan (height + 1)))).expectation
(fun direction => (PreliminarySampler.questionsLaw (branch := branch parameters) (n := height + 1)
(t := sourceLength parameters.plan hδ) (m := m)).expectation (fun q =>
(HierarchicalAllDecoderTables.upperTableLaw
(SourceQuestionLowerForms.slots ((Nodes.path upper).append pDown) outside clauses q)
upper (height + 1) σ (useful q) adviceRows density).expectation (fun table =>
(tagLaw parameters).expectation (fun tag =>
SourceQuestionPhysicalHigh.collision ((Nodes.path upper).append pDown) outside placeholder
clauses designated q tag.1 tag.2 upper (height + 1) d
(FixedLowerRawCollision.branchPositive parameters) A exterior direction
(cutoff parameters.plan hδ) table)))) := by
unfold probability
apply (CollisionAveraging.mean_square_le _ _).trans
apply SmallBias.expectation_mono
intro direction
apply (CollisionAveraging.mean_square_le _ _).trans
apply SmallBias.expectation_mono
intro q
apply (CollisionAveraging.mean_square_le _ _).trans
rw [FiniteDistribution.expectation_comm]
apply SmallBias.expectation_mono
intro tag
exact SourceQuestionPhysicalHigh.success_square_le outside placeholder clauses designated q tag.1 tag.2
upper pDown hproper (height + 1) (FixedLowerRawCollision.branchPositive parameters) d hnode A cut
σ (useful q) density threshold exterior direction (cutoff parameters.plan hδ)
end
end PerfectCompleteness.FixedSourcePhysicalHigh
end
section
namespace PerfectCompleteness.FixedSourceHighBound
noncomputable section
open scoped Classical BigOperators
open FixedParameters FixedRows RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays
open UniqueGamesTheorem.Foundations.Games
variable {δ : ℚ} {hδ : 0 < δ} (parameters : Parameters δ hδ)
{height v m : Nat} [NeZero m]
(outside : Slots (branch parameters) parameters.plan.depth →
Fin (sourceLength parameters.plan hδ) → MixedSupport.Slot)
(placeholder : Slots (branch parameters) (height + 1) →
Fin (sourceLength parameters.plan hδ) → MixedSupport.Slot)
(clauses : Fin m → SourceClause.NormalizedClause v)
(designated : Fin (branch parameters height) → Slots (branch parameters) height)
(upper : Nodes (branch parameters) parameters.plan.depth)
(pDown : Path (branch parameters) (Nodes.height upper) (height + 1))
(hproper : height + 1 < Nodes.height upper)
(d : HierarchicalFrozenTables.LowerNodes upper (height + 1))
(hnode : WholeArrayInteriorExterior.upperNode ((Nodes.path upper).append pDown) =
HierarchicalLeftDecoder.LowerNode upper (height + 1) d)
{adviceRows : Nat}
(A : ManyGoodRows.RowMap (Block (rows parameters.plan) upper) adviceRows)
(cut : OwnInputReference.Cut upper (ProjectedLowerFiber.lower upper (height + 1) d))
(exterior : CleanPhysicalReplay.Exterior (rows parameters.plan) (repeats parameters.plan)
((Nodes.path upper).append pDown) outside placeholder)
(σ : KeyStrategy.Strategy (TreeCanonical.locationCount (branch parameters) parameters.plan.depth
(sourceLength parameters.plan hδ)))
(useful : (q : FixedSourceGlobalCollision.Questions (m := m) parameters (height := height)) →
(bg : HierarchicalMatrixTable.Background (rows := rows parameters.plan)
(SourceQuestionLowerForms.slots ((Nodes.path upper).append pDown) outside clauses q) upper) →
HierarchicalFrozenTables.QuotientMatrix
(SourceQuestionLowerForms.slots ((Nodes.path upper).append pDown) outside clauses q)
upper (height + 1) bg → Prop)
(density threshold : ℝ)
local instance rowSpaceFintype (q : FixedSourceGlobalCollision.Questions
(m := m) parameters (height := height)) :
Fintype (NodeEmbedding.RowSpace
(SourceQuestionLowerForms.slots ((Nodes.path upper).append pDown) outside clauses q) upper) :=
Fintype.ofFinite _
local instance upperAnswerFintype (q : FixedSourceGlobalCollision.Questions
(m := m) parameters (height := height)) :
Fintype (HierarchicalAllDecoderTables.UpperAnswer
(SourceQuestionLowerForms.slots ((Nodes.path upper).append pDown) outside clauses q) upper) :=
LeftDecoder.dualFintype
(V := NodeEmbedding.RowSpace
(SourceQuestionLowerForms.slots ((Nodes.path upper).append pDown) outside clauses q) upper)
local instance directionNonempty :
Nonempty (BucketSampler.Direction (rows parameters.plan (height + 1))) :=
⟨BucketUniform.coordinateDirection ⟨0, lt_of_lt_of_le Nat.zero_lt_one
(parameters.plan.rows_pos (parameters.plan.depth - (height + 1)))⟩⟩
omit [NeZero m] in
theorem tag_collision_eq
(direction : BucketSampler.Direction (rows parameters.plan (height + 1)))
(q : FixedSourceGlobalCollision.Questions (m := m) parameters (height := height))
(table : SourceQuestionLowerForms.UpperTable (rows := rows parameters.plan)
((Nodes.path upper).append pDown) outside clauses upper (height + 1)
(adviceRows := adviceRows) q) :
(FixedSourcePhysicalHigh.tagLaw parameters).expectation (fun tag =>
SourceQuestionPhysicalHigh.collision ((Nodes.path upper).append pDown) outside placeholder
clauses designated q tag.1 tag.2 upper (height + 1) d
(FixedLowerRawCollision.branchPositive parameters) A exterior direction
(cutoff parameters.plan hδ) table) =
FixedSourceGlobalCollision.localProbability parameters ((Nodes.path upper).append pDown)
outside placeholder clauses designated upper d A exterior q table direction := by
simpa only [FixedSourcePhysicalHigh.tagLaw, SourceProjectedTag.law,
FiniteDistribution.expectation_product, FixedSourceGlobalCollision.localProbability,
FixedSourceGlobalCollision.localTables, Function.update_self] using
(SourceQuestionPhysicalCollision.mean_collision_eq ((Nodes.path upper).append pDown)
outside placeholder clauses designated upper (height + 1) d
(FixedLowerRawCollision.branchPositive parameters) A exterior q direction
(cutoff parameters.plan hδ)
(FixedSourceGlobalCollision.localTables parameters ((Nodes.path upper).append pDown)
outside clauses upper q table) (FixedSourceGlobalCollision.flag parameters))
theorem probability_square_le (hdensity : 0 < density) :
FixedSourcePhysicalHigh.probability parameters outside placeholder clauses designated upper pDown
hproper d hnode A cut exterior σ useful density threshold ^ 2 ≤
FixedRankContradiction.gamma parameters (Nodes.height upper) ^ 2 / 8 +
2 * parameters.accuracy := by
have h := FixedSourcePhysicalHigh.probability_square_le_collision parameters outside placeholder
clauses designated upper pDown hproper d hnode A cut exterior σ useful density threshold
simp_rw [tag_collision_eq] at h
exact h.trans (FixedSourceGlobalCollision.mean_local_le parameters ((Nodes.path upper).append pDown)
outside placeholder clauses designated upper d hnode A exterior σ useful density hdensity)
end
end PerfectCompleteness.FixedSourceHighBound
end
end OAI
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