File size: 7,441 Bytes
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 | import OAI.Computability.PerfectCompleteness.Construction.HierarchicalAdviceFromBuckets
import OAI.Computability.PerfectCompleteness.Construction.StoppedProjectedBucketIndex
import OAI.Computability.PerfectCompleteness.Foundations.StoppedPathHead
import OAI.Computability.PerfectCompleteness.Sampling.StoppedUsefulPairLaw
namespace OAI
section
namespace PerfectCompleteness.StoppedProjectedBuckets
noncomputable section
open scoped Classical
open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays
open UniqueGamesTheorem.Foundations.Games
open UniqueGamesTheorem.Appendix.RankLevelFilter (linearMapFintype)
attribute [local instance] linearMapFintype
variable {branch : Nat → Nat} {n i j t v m : Nat}
(clauses : Fin m → SourceClause.NormalizedClause v)
(rows repeats : Nat → Nat) (hupper : j + 1 ≤ n) (hij : i < j)
(designated : Fin (branch i) → Slots branch i)
(hrows : ∀ k, 0 < rows (k + 1))
(o : StoppedProjectedExperiment.Outer
(branch := branch) (n := n) (j := j) (t := t) (m := m))
abbrev Base := StoppedProjectedBucketIndex.Base (branch := branch) (i := i) (j := j) (t := t)
abbrev key (base : Base (branch := branch) (i := i) (j := j) (t := t)) :=
StoppedProjectedBucketIndex.key (n := n) rows hrows base
abbrev lowerPath (base : Base (branch := branch) (i := i) (j := j) (t := t)) :=
StoppedProjectedExperiment.lowerPath rows hij (key (n := n) rows hrows base)
def chosen (base : Base (branch := branch) (i := i) (j := j) (t := t)) : Fin (branch j) :=
StoppedPathHead.child hij (lowerPath (n := n) rows hij hrows base)
def continuation (base : Base (branch := branch) (i := i) (j := j) (t := t)) :
Path branch j (i + 1) :=
StoppedPathHead.tail hij (lowerPath (n := n) rows hij hrows base)
theorem lowerPath_eq_step (base : Base (branch := branch) (i := i) (j := j) (t := t)) :
lowerPath (n := n) rows hij hrows base =
.step (chosen (n := n) rows hij hrows base) (continuation (n := n) rows hij hrows base) :=
StoppedPathHead.step_eq hij _
abbrev projected (base : Base (branch := branch) (i := i) (j := j) (t := t)) :=
StoppedProjectedExperiment.projectedSlots clauses rows hupper hij designated o
(key (n := n) rows hrows base)
abbrev projection (base : Base (branch := branch) (i := i) (j := j) (t := t)) :=
StoppedProjectedExperiment.projection clauses rows hupper hij designated o
(key (n := n) rows hrows base)
abbrev Exterior (base : Base (branch := branch) (i := i) (j := j) (t := t)) :=
StoppedUsefulPairLaw.Descriptor rows repeats (StoppedProjectedExperiment.upperPath hupper o)
(chosen (n := n) rows hij hrows base) (continuation (n := n) rows hij hrows base)
(projected clauses rows hupper hij designated hrows o base)
abbrev Descriptor := (base : Base (branch := branch) (i := i) (j := j) (t := t)) ×
Exterior clauses rows repeats hupper hij designated hrows o base
def exteriorLaw (base : Base (branch := branch) (i := i) (j := j) (t := t)) :
FiniteDistribution (Exterior clauses rows repeats hupper hij designated hrows o base) :=
StoppedUsefulPairLaw.externalLaw rows repeats (StoppedProjectedExperiment.upperPath hupper o)
(chosen (n := n) rows hij hrows base) (continuation (n := n) rows hij hrows base)
(projected clauses rows hupper hij designated hrows o base)
def externalLaw (hbranch : ∀ k < j + 1, 0 < branch k)
(flag : Fin (branch i) → FiniteDistribution Bool) :
FiniteDistribution (Descriptor clauses rows repeats hupper hij designated hrows o) :=
CompletionSoundness.sigmaLaw
(StoppedProjectedBucketIndex.baseLaw (t := t) hij hbranch flag)
(exteriorLaw clauses rows repeats hupper hij designated hrows o)
local instance backgroundFintype : Fintype (HierarchicalMatrixTable.Background (rows := rows)
(StoppedProjectedExperiment.nativeSlots clauses o) (StoppedProjectedExperiment.upper hupper o)) :=
Fintype.ofFinite _
local instance rowSpaceFintype : Fintype (NodeEmbedding.RowSpace
(StoppedProjectedExperiment.nativeSlots clauses o) (StoppedProjectedExperiment.upper hupper o)) :=
Fintype.ofFinite _
def background (descriptor : Descriptor clauses rows repeats hupper hij designated hrows o) :
HierarchicalMatrixTable.Background (rows := rows)
(StoppedProjectedExperiment.nativeSlots clauses o) (StoppedProjectedExperiment.upper hupper o) :=
StoppedUsefulPairLaw.background rows repeats (StoppedProjectedExperiment.upperPath hupper o)
(chosen (n := n) rows hij hrows descriptor.1)
(continuation (n := n) rows hij hrows descriptor.1)
(StoppedProjectedExperiment.nativeSlots clauses o)
(projected clauses rows hupper hij designated hrows o descriptor.1)
(projection clauses rows hupper hij designated hrows o descriptor.1) descriptor.2
def scalarLaw (descriptor : Descriptor clauses rows repeats hupper hij designated hrows o) :
FiniteDistribution (NodeEmbedding.RowSpace
(StoppedProjectedExperiment.nativeSlots clauses o) (StoppedProjectedExperiment.upper hupper o)) :=
StoppedUsefulPairLaw.scalarLaw rows repeats (StoppedProjectedExperiment.upperPath hupper o)
(chosen (n := n) rows hij hrows descriptor.1)
(continuation (n := n) rows hij hrows descriptor.1)
(StoppedProjectedExperiment.nativeSlots clauses o)
(projected clauses rows hupper hij designated hrows o descriptor.1)
(projection clauses rows hupper hij designated hrows o descriptor.1) descriptor.2
variable (hbranch : ∀ k < j + 1, 0 < branch k)
(flag : Fin (branch i) → FiniteDistribution Bool)
(σ : KeyStrategy.Strategy (TreeCanonical.locationCount branch n t))
abbrev experiment :=
StoppedProjectedExperiment.experiment clauses rows repeats hupper hij designated
hbranch hrows flag σ o
include hrows in
theorem rows_positive : 0 < rows (Nodes.height (StoppedProjectedExperiment.upper hupper o)) := by
rw [StoppedProjectedExperiment.upper_height]
exact hrows j
def usefulProbability (κ : ℝ) : ℝ :=
HierarchicalAdviceFromBuckets.usefulProbability
(experiment clauses rows repeats hupper hij designated hrows o hbranch flag σ) κ
(externalLaw clauses rows repeats hupper hij designated hrows o hbranch flag)
(background clauses rows repeats hupper hij designated hrows o)
(scalarLaw clauses rows repeats hupper hij designated hrows o)
(rows_positive rows hupper hrows o)
theorem usefulProbability_eq_mean (κ : ℝ) :
usefulProbability clauses rows repeats hupper hij designated hrows o hbranch flag σ κ =
(StoppedProjectedBucketIndex.baseLaw (t := t) hij hbranch flag).expectation (fun base =>
HierarchicalAdviceFromBuckets.usefulProbability
(experiment clauses rows repeats hupper hij designated hrows o hbranch flag σ) κ
(exteriorLaw clauses rows repeats hupper hij designated hrows o base)
(fun external => background clauses rows repeats hupper hij designated hrows o
⟨base, external⟩)
(fun external => scalarLaw clauses rows repeats hupper hij designated hrows o
⟨base, external⟩)
(rows_positive rows hupper hrows o)) := by
simp only [usefulProbability, HierarchicalAdviceFromBuckets.usefulProbability,
HierarchicalUsefulCollision.law, CompletionSoundness.sigmaLaw_probability,
externalLaw, CandidateCoupling.expectation_sigmaLaw,
HierarchicalUsefulCollision.usefulEvent]
end
end PerfectCompleteness.StoppedProjectedBuckets
end
end OAI
|