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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 | import OAI.Computability.PerfectCompleteness.Decoding.FixedDecoderCleanRateLemmas
import OAI.Computability.PerfectCompleteness.Reduction.FixedSourcePhysicalHighLemmas
import OAI.Computability.PerfectCompleteness.Repetition.CleanPhysicalHighAverage
namespace OAI
section
namespace PerfectCompleteness.FixedNativeHigh
noncomputable section
open scoped Classical
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]
{upper : Nodes (branch parameters) parameters.plan.depth}
{d : HierarchicalFrozenTables.LowerNodes upper (height + 1)}
(S : CleanDecoderRate.Setup (branch parameters) (rows parameters.plan) (repeats parameters.plan)
parameters.plan.depth height (sourceLength parameters.plan hδ) v m upper d)
(hnode : WholeArrayInteriorExterior.upperNode (CleanDecoderRate.path S) =
HierarchicalLeftDecoder.LowerNode upper (height + 1) d)
{Original : OriginalDecoderMark.SlotFamily (branch parameters) parameters.plan.depth
(sourceLength parameters.plan hδ) → Type*}
[∀ slots, Fintype (Original slots)]
(F : OriginalDecoderMark.Family (branch parameters) (rows parameters.plan)
parameters.plan.depth (sourceLength parameters.plan hδ) Original)
(κ : ℝ)
(σ : KeyStrategy.Strategy (TreeCanonical.locationCount (branch parameters) parameters.plan.depth
(sourceLength parameters.plan hδ)))
(r : Nat) (density : ℝ)
(A : ManyGoodRows.RowMap (Block (rows parameters.plan) upper) r)
(threshold : ℝ)
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)))⟩⟩
theorem flag_eq (h : Nat) :
FixedSourceGlobalCollision.flag parameters (height := h) =
fun _ : Fin (branch parameters h) =>
ProjectedCleanRate.projectionFlag (parameters.cubeSize h)
(FixedDecoderCleanRate.cubePositive parameters h) := by
funext child
apply FiniteDistribution.eq_of_weight_eq
intro b
cases b <;>
simp [FixedSourceGlobalCollision.flag, ProjectedCleanRate.projectionFlag,
ProjectionPosterior.bernoulli, RepetitionRate.bernoulli,
FixedParameters.projectionProbability, ProjectedCleanRate.projectionDensity]
def questionUseful
(q : FixedSourceGlobalCollision.Questions (m := m) parameters (height := height)) :=
OriginalDecoderMark.mark F upper (height + 1) κ
(SourceQuestionLowerForms.slots (CleanDecoderRate.path S) S.outside S.clauses q)
def probability : ℝ :=
(FiniteDistribution.uniform (BucketSampler.Direction (rows parameters.plan (height + 1)))).expectation
(fun direction =>
(CleanDecoderPairLaw.pairLaw S σ (CleanPhysicalDecoderLaw.useful S F κ)
r density A (LinearMap.ker A) (CleanPhysicalHighAverage.nativeDirection S hnode direction)
threshold (parameters.cubeSize height) (FixedDecoderCleanRate.cubePositive parameters height)).probability
(fun z => DecoderRankSplit.highAgrees S (FixedLowerRawCollision.branchPositive parameters)
(cutoff parameters.plan hδ) z.1 z.2))
theorem probability_eq_physical :
probability parameters S hnode F κ σ r density A threshold =
FixedSourcePhysicalHigh.probability parameters S.outside S.placeholder S.clauses S.designated
upper (CleanPhysicalHighAverage.suffix S) (CleanPhysicalHighAverage.suffix_proper S)
d hnode A S.cut S.exterior σ (questionUseful parameters S F κ) density threshold := by
unfold probability FixedSourcePhysicalHigh.probability
apply FiniteDistribution.expectation_congr
intro direction
rw [CleanPhysicalHighAverage.probability_eq_tag_average S
(FixedLowerRawCollision.branchPositive parameters) hnode F κ σ r density A direction
threshold (cutoff parameters.plan hδ) (parameters.cubeSize height)
(FixedDecoderCleanRate.cubePositive parameters height)]
simp only [FixedSourcePhysicalHigh.tagLaw, flag_eq]
apply FiniteDistribution.expectation_congr
intro q
apply FiniteDistribution.expectation_congr
intro tag
rfl
theorem probability_square_le (hdensity : 0 < density) :
probability parameters S hnode F κ σ r density A threshold ^ 2 ≤
FixedRankContradiction.gamma parameters (Nodes.height upper) ^ 2 / 8 +
2 * parameters.accuracy := by
rw [probability_eq_physical]
exact FixedSourceHighBound.probability_square_le parameters S.outside S.placeholder S.clauses
S.designated upper (CleanPhysicalHighAverage.suffix S) (CleanPhysicalHighAverage.suffix_proper S)
d hnode A S.cut S.exterior σ (questionUseful parameters S F κ) density threshold hdensity
end
end PerfectCompleteness.FixedNativeHigh
end
end OAI
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