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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 | import OAI.Computability.PerfectCompleteness.Decoding.FixedSourceChildCollision
import OAI.Computability.PerfectCompleteness.Decoding.LowerCutScalarProjectionLemmas
import OAI.Computability.PerfectCompleteness.Decoding.LowerCutUniformCoordinatesLemmas
namespace OAI
section
namespace PerfectCompleteness.FixedSourceLowerCollision
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]
(path : Path (branch parameters) parameters.plan.depth (height + 1))
(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)
(d : HierarchicalFrozenTables.LowerNodes upper (height + 1))
(hnode : WholeArrayInteriorExterior.upperNode path =
HierarchicalLeftDecoder.LowerNode upper (height + 1) d)
{adviceRows : Nat}
(A : ManyGoodRows.RowMap (Block (rows parameters.plan) upper) adviceRows)
(exterior : CleanPhysicalReplay.Exterior (rows parameters.plan) (repeats parameters.plan)
path outside placeholder)
abbrev Questions := FixedSourceChildCollision.Questions
(m := m) parameters (height := height)
abbrev Table (q : Questions (m := m) parameters (height := height)) :=
SourceQuestionLowerForms.UpperTable (rows := rows parameters.plan)
path outside clauses upper (height + 1) (adviceRows := adviceRows) q
local instance rowSpaceFintype (q : Questions (m := m) parameters (height := height)) :
Fintype (NodeEmbedding.RowSpace (SourceQuestionLowerForms.slots path outside clauses q) upper) :=
Fintype.ofFinite _
local instance upperAnswerFintype (q : Questions (m := m) parameters (height := height)) :
Fintype (HierarchicalAllDecoderTables.UpperAnswer
(SourceQuestionLowerForms.slots path outside clauses q) upper) :=
LeftDecoder.dualFintype
(V := NodeEmbedding.RowSpace (SourceQuestionLowerForms.slots path outside clauses q) upper)
local instance tableFintype (q : Questions (m := m) parameters (height := height)) :
Fintype (Table parameters path outside clauses upper (adviceRows := adviceRows) q) :=
inferInstanceAs (Fintype (HierarchicalAllDecoderTables.Input (rows := rows parameters.plan)
(SourceQuestionLowerForms.slots path outside clauses q) upper (height + 1) adviceRows →
HierarchicalAllDecoderTables.UpperAnswer
(SourceQuestionLowerForms.slots path outside clauses q) upper))
abbrev TableFamily := (q : Questions (m := m) parameters (height := height)) →
Table parameters path outside clauses upper (adviceRows := adviceRows) q
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 first (tables : TableFamily parameters path outside clauses upper (adviceRows := adviceRows)) :=
SourceQuestionLowerForms.first path outside placeholder clauses upper (height + 1) d hnode
(FixedLowerRawCollision.branchPositive parameters) A exterior tables (cutoff parameters.plan hδ)
def second (tables : TableFamily parameters path outside clauses upper (adviceRows := adviceRows))
(direction : BucketSampler.Direction (rows parameters.plan (height + 1))) :=
SourceQuestionLowerForms.second path outside placeholder clauses upper (height + 1) d hnode
(FixedLowerRawCollision.branchPositive parameters) A exterior tables (cutoff parameters.plan hδ)
direction
def markedProbability
(tables : TableFamily parameters path outside clauses upper (adviceRows := adviceRows))
(direction : BucketSampler.Direction (rows parameters.plan (height + 1))) : ℝ :=
(FixedSourceChildCollision.markedLaw parameters path clauses designated).probability
(FixedSourceChildCollision.restrictedEvent parameters path clauses designated
(first parameters path outside placeholder clauses upper d hnode A exterior tables)
(second parameters path outside placeholder clauses upper d hnode A exterior tables direction))
def uniformProbability
(tables : TableFamily parameters path outside clauses upper (adviceRows := adviceRows))
(direction : BucketSampler.Direction (rows parameters.plan (height + 1))) : ℝ :=
(FixedSourceChildCollision.uniformLaw parameters path clauses).probability
(FixedSourceChildCollision.fullEvent parameters path clauses
(first parameters path outside placeholder clauses upper d hnode A exterior tables)
(second parameters path outside placeholder clauses upper d hnode A exterior tables direction))
variable
(σ : KeyStrategy.Strategy (TreeCanonical.locationCount (branch parameters) parameters.plan.depth
(sourceLength parameters.plan hδ)))
(useful : (q : Questions (m := m) parameters (height := height)) →
(bg : HierarchicalMatrixTable.Background (rows := rows parameters.plan)
(SourceQuestionLowerForms.slots path outside clauses q) upper) →
HierarchicalFrozenTables.QuotientMatrix
(SourceQuestionLowerForms.slots path outside clauses q) upper (height + 1) bg → Prop)
(density : ℝ)
def tableLaw : FiniteDistribution
(TableFamily parameters path outside clauses upper (adviceRows := adviceRows)) :=
FiniteProduct.law (fun q => HierarchicalAllDecoderTables.upperTableLaw
(SourceQuestionLowerForms.slots path outside clauses q) upper (height + 1)
σ (useful q) adviceRows density)
omit [NeZero m] in
private theorem supported_row
(tables : TableFamily parameters path outside clauses upper (adviceRows := adviceRows))
(hsupport : (tableLaw parameters path outside clauses upper σ useful density).weight tables ≠ 0)
(q : Questions (m := m) parameters (height := height)) :
(HierarchicalAllDecoderTables.upperTableLaw
(SourceQuestionLowerForms.slots path outside clauses q) upper (height + 1)
σ (useful q) adviceRows density).weight (tables q) ≠ 0 := by
intro hz
apply hsupport
exact Finset.prod_eq_zero (Finset.mem_univ q) hz
theorem supported_uniform_le (hdensity : 0 < density)
(tables : TableFamily parameters path outside clauses upper (adviceRows := adviceRows))
(hsupport : (tableLaw parameters path outside clauses upper σ useful density).weight tables ≠ 0) :
(FiniteDistribution.uniform (BucketSampler.Direction (rows parameters.plan (height + 1)))).expectation
(uniformProbability parameters path outside placeholder clauses upper d hnode A exterior tables) ≤
FixedRankContradiction.gamma parameters (Nodes.height upper) ^ 2 / 8 := by
have hprob :
uniformProbability parameters path outside placeholder clauses upper d hnode A exterior tables =
fun direction => (PreliminarySampler.questionsLaw
(branch := branch parameters) (n := height + 1)
(t := sourceLength parameters.plan hδ) (m := m)).expectation (fun q =>
(CutChildGrouping.rawLaw (C := FixedSourceChildCollision.Calls parameters path)
(SourceChildMarkedLaw.questionSlots clauses q) (rows parameters.plan)).probability
(fun raw => BilinearCollisionTransfer.fullCollision
(first parameters path outside placeholder clauses upper d hnode A exterior tables q raw)
(second parameters path outside placeholder clauses upper d hnode A exterior tables
direction q raw))) := by
funext direction
exact CompletionSoundness.sigmaLaw_probability _ _ _
rw [hprob, FiniteDistribution.expectation_comm]
calc
_ ≤ (PreliminarySampler.questionsLaw (branch := branch parameters) (n := height + 1)
(t := sourceLength parameters.plan hδ) (m := m)).expectation
(fun _ => FixedRankContradiction.gamma parameters (Nodes.height upper) ^ 2 / 8) := by
apply SmallBias.expectation_mono
intro q
have heq :
(FiniteDistribution.uniform (BucketSampler.Direction
(rows parameters.plan (height + 1)))).expectation
(fun direction => (CutChildGrouping.rawLaw
(C := FixedSourceChildCollision.Calls parameters path)
(SourceChildMarkedLaw.questionSlots clauses q) (rows parameters.plan)).probability
(fun raw => BilinearCollisionTransfer.fullCollision
(first parameters path outside placeholder clauses upper d hnode A exterior tables q raw)
(second parameters path outside placeholder clauses upper d hnode A exterior tables
direction q raw))) =
FixedLowerRawCollision.fullCollisionProbability parameters path
(SourceQuestionLowerForms.slots path outside clauses q) upper d hnode A
(CleanPhysicalReplay.exteriorAt (rows parameters.plan) (repeats parameters.plan)
path outside placeholder (SourceChildMarkedLaw.questionSlots clauses q) exterior)
(tables q) := by
apply FiniteDistribution.expectation_congr
intro direction
exact SourceQuestionLowerForms.fullCollision_probability path outside placeholder clauses upper
(height + 1) d hnode (FixedLowerRawCollision.branchPositive parameters) A exterior tables
(cutoff parameters.plan hδ) direction q
rw [heq]
exact FixedLowerRawCollision.supported_fullCollision_le parameters path
(SourceQuestionLowerForms.slots path outside clauses q) upper d hnode A
(CleanPhysicalReplay.exteriorAt (rows parameters.plan) (repeats parameters.plan)
path outside placeholder (SourceChildMarkedLaw.questionSlots clauses q) exterior)
σ (useful q) density hdensity (tables q)
(supported_row parameters path outside clauses upper σ useful density tables hsupport q)
_ = _ := SmallBias.expectation_const _ _
private theorem expectation_add_const {X : Type*} [Fintype X]
(μ : FiniteDistribution X) (f : X → ℝ) (c : ℝ) :
μ.expectation (fun x => f x + c) = μ.expectation f + c := by
simp only [FiniteDistribution.expectation, mul_add, Finset.sum_add_distrib, ← Finset.sum_mul, μ.normalized, one_mul]
theorem mean_marked_le (hdensity : 0 < density) :
(tableLaw parameters path outside clauses upper σ useful density).expectation
(fun tables => (FiniteDistribution.uniform
(BucketSampler.Direction (rows parameters.plan (height + 1)))).expectation
(markedProbability parameters path outside placeholder clauses designated upper d hnode
A exterior tables)) ≤
FixedRankContradiction.gamma parameters (Nodes.height upper) ^ 2 / 8 +
2 * parameters.accuracy := by
apply DecoderTableAdmissibility.expectation_le_of_support
intro tables hsupport
have hpoint : ∀ direction : BucketSampler.Direction (rows parameters.plan (height + 1)),
markedProbability parameters path outside placeholder clauses designated upper d hnode
A exterior tables direction ≤
uniformProbability parameters path outside placeholder clauses upper d hnode
A exterior tables direction + 2 * parameters.accuracy := by
intro direction
exact FixedSourceChildCollision.restricted_probability_le parameters path clauses designated
(first parameters path outside placeholder clauses upper d hnode A exterior tables)
(second parameters path outside placeholder clauses upper d hnode A exterior tables direction)
le_rfl
have hmean := SmallBias.expectation_mono
(FiniteDistribution.uniform (BucketSampler.Direction (rows parameters.plan (height + 1)))) hpoint
rw [expectation_add_const] at hmean
have huniform := supported_uniform_le parameters path outside placeholder clauses upper d hnode A exterior
σ useful density hdensity tables hsupport
linarith
end
end PerfectCompleteness.FixedSourceLowerCollision
end
end OAI
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