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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 | import OAI.Computability.PerfectCompleteness.Construction.SourcePhysicalTaggedChildren
import OAI.Computability.PerfectCompleteness.Decoding.CutProjectionAssembly
import OAI.Computability.PerfectCompleteness.Decoding.LowerCutScalarProjectionLemmas
import OAI.Computability.PerfectCompleteness.Decoding.LowerProjectedOriginalSuccessLemmas
import OAI.Computability.PerfectCompleteness.Foundations.SourceProjectedTag
namespace OAI
section
namespace PerfectCompleteness.SourceQuestionPhysicalHigh
noncomputable section
open scoped Classical
open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays
open UniqueGamesTheorem.Foundations.Games
attribute [local instance] RightDecoder.scalarFintype
variable {branch rows repeats : Nat → Nat} {n height t v m : Nat}
(path : Path branch n (height + 1))
(outside : Slots branch n → Fin t → MixedSupport.Slot)
(placeholder : Slots branch (height + 1) → Fin t → MixedSupport.Slot)
(clauses : Fin m → SourceClause.NormalizedClause v)
(designated : Fin (branch height) → Slots branch height)
(q : SourceQuestionLowerForms.Questions
(branch := branch) (height := height) (t := t) (m := m))
(choices : SourceQuestionKernelJoint.ChoiceTuple
(branch := branch) (n := height) (t := t))
(flags : SourceProjectedTag.Flags (branch := branch) (n := height))
abbrev originalSlots := SourceQuestionLowerForms.slots path outside clauses q
def projectedSlots : Slots branch n → Fin t → MixedSupport.Slot :=
CutSlotAssembly.fill path outside
(SourceProjectedTag.mixedInside clauses designated q choices flags)
def projection : ∀ leaf j, MixedSupport.Projection
(originalSlots path outside clauses q leaf j)
(projectedSlots path outside clauses designated q choices flags leaf j) :=
CutProjectionAssembly.fillProjection path outside
(SourceChildMarkedLaw.questionSlots clauses q)
(SourceProjectedTag.mixedInside clauses designated q choices flags)
(SourceProjectedTag.projection clauses designated q choices flags)
def exteriorAt
(exterior : CleanPhysicalReplay.Exterior rows repeats path outside placeholder) :
WholeCutGrouping.Exterior rows repeats path
(projectedSlots path outside clauses designated q choices flags) :=
CleanPhysicalReplay.exteriorAt rows repeats path outside placeholder
(SourceProjectedTag.mixedInside clauses designated q choices flags) exterior
section Collision
variable (upper : Nodes branch n) (level : Nat)
(d : HierarchicalFrozenTables.LowerNodes upper level)
(hbranch : ∀ k < n, 0 < branch k)
{adviceRows : Nat} (A : ManyGoodRows.RowMap (Block rows upper) adviceRows)
(exterior : CleanPhysicalReplay.Exterior rows repeats path outside placeholder)
(direction : BucketSampler.Direction (rows (height + 1)))
(cutoff : Nat)
(table : SourceQuestionLowerForms.UpperTable (rows := rows)
path outside clauses upper level (adviceRows := adviceRows) q)
def collision : ℝ :=
LowerProjectedFiberAverage.rawCollision path
(originalSlots path outside clauses q)
(projectedSlots path outside clauses designated q choices flags)
(projection path outside clauses designated q choices flags)
upper level hbranch d direction
(exteriorAt path outside placeholder clauses designated q choices flags exterior)
A cutoff table
end Collision
section Success
variable (upper : Nodes branch n)
(pDown : Path branch (Nodes.height upper) (height + 1))
(hproper : height + 1 < Nodes.height upper)
(level : Nat) (hbranch : ∀ k < n, 0 < branch k)
(d : HierarchicalFrozenTables.LowerNodes upper level)
(hnode : WholeArrayInteriorExterior.upperNode ((Nodes.path upper).append pDown) =
HierarchicalLeftDecoder.LowerNode upper level d)
{adviceRows : Nat} (A : ManyGoodRows.RowMap (Block rows upper) adviceRows)
(cut : OwnInputReference.Cut upper (ProjectedLowerFiber.lower upper level d))
(σ : KeyStrategy.Strategy (TreeCanonical.locationCount branch n t))
(useful : (bg : HierarchicalMatrixTable.Background (rows := rows)
(originalSlots ((Nodes.path upper).append pDown) outside clauses q) upper) →
HierarchicalFrozenTables.QuotientMatrix
(originalSlots ((Nodes.path upper).append pDown) outside clauses q) upper level bg → Prop)
(ρ threshold : ℝ)
(exterior : CleanPhysicalReplay.Exterior rows repeats
((Nodes.path upper).append pDown) outside placeholder)
(direction : BucketSampler.Direction (rows (height + 1)))
(cutoff : Nat)
local instance rowSpaceFintype :
Fintype (NodeEmbedding.RowSpace
(originalSlots ((Nodes.path upper).append pDown) outside clauses q) upper) :=
Fintype.ofFinite _
local instance upperAnswerFintype :
Fintype (HierarchicalAllDecoderTables.UpperAnswer
(originalSlots ((Nodes.path upper).append pDown) outside clauses q) upper) :=
LeftDecoder.dualFintype (V := NodeEmbedding.RowSpace
(originalSlots ((Nodes.path upper).append pDown) outside clauses q) upper)
def success : ℝ :=
LowerPhysicalFiberSuccess.physicalSuccess
(originalSlots ((Nodes.path upper).append pDown) outside clauses q)
(projectedSlots ((Nodes.path upper).append pDown) outside clauses designated q choices flags)
(projection ((Nodes.path upper).append pDown) outside clauses designated q choices flags)
upper pDown hproper level hbranch d hnode direction
(exteriorAt ((Nodes.path upper).append pDown) outside placeholder clauses designated
q choices flags exterior) A cut σ useful ρ threshold cutoff
theorem success_square_le :
success outside placeholder clauses designated q choices flags upper pDown hproper level
hbranch d hnode A cut σ useful ρ threshold exterior direction cutoff ^ 2 ≤
(HierarchicalAllDecoderTables.upperTableLaw
(originalSlots ((Nodes.path upper).append pDown) outside clauses q)
upper level σ useful adviceRows ρ).expectation
(collision ((Nodes.path upper).append pDown) outside placeholder clauses designated
q choices flags upper level d hbranch A exterior direction cutoff) :=
LowerPhysicalFiberSuccess.physicalSuccess_square_le
(originalSlots ((Nodes.path upper).append pDown) outside clauses q)
(projectedSlots ((Nodes.path upper).append pDown) outside clauses designated q choices flags)
(projection ((Nodes.path upper).append pDown) outside clauses designated q choices flags)
upper pDown hproper level hbranch d hnode direction
(exteriorAt ((Nodes.path upper).append pDown) outside placeholder clauses designated
q choices flags exterior) A cut σ useful ρ threshold cutoff
end Success
end
end PerfectCompleteness.SourceQuestionPhysicalHigh
end
end OAI
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