File size: 15,981 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 | import OAI.Computability.PerfectCompleteness.Construction.SourcePhysicalChildExpectation
import OAI.Computability.PerfectCompleteness.Construction.SourceQuestionPhysicalHigh
import OAI.Computability.PerfectCompleteness.Repetition.CleanPhysicalHighEvent
import OAI.Computability.PerfectCompleteness.Repetition.CleanPhysicalRawChildren
namespace OAI
section
namespace PerfectCompleteness.CleanPhysicalHighAverage
noncomputable section
open scoped Classical
open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays
open UniqueGamesTheorem.Foundations.Games
private theorem mpr_heq {α β : Sort _} (h : α = β) (x : β) :
HEq (Eq.mpr h x) x := by
cases h
rfl
variable {branch rows repeats : Nat → Nat} {n h t v m : Nat} [NeZero m]
{upper : Nodes branch n} {d : HierarchicalFrozenTables.LowerNodes upper (h + 1)}
(S : CleanDecoderRate.Setup branch rows repeats n h t v m upper d)
abbrev suffix := CleanRightDecoder.suffix upper (CleanPhysicalDecoderLaw.lower S)
S.cut (CleanDecoderContext.lowerHeight upper d)
include S in
theorem suffix_proper : h + 1 < Nodes.height upper :=
CleanRightDecoder.suffix_proper upper (CleanPhysicalDecoderLaw.lower S)
S.cut (CleanDecoderContext.lowerHeight upper d)
variable (hbranch : ∀ k < n, 0 < branch k)
(hnode : WholeArrayInteriorExterior.upperNode (CleanDecoderRate.path S) =
HierarchicalLeftDecoder.LowerNode upper (h + 1) d)
{Original : OriginalDecoderMark.SlotFamily branch n t → Type*}
[∀ slots, Fintype (Original slots)]
(F : OriginalDecoderMark.Family branch rows n t Original) (κ : ℝ)
(σ : KeyStrategy.Strategy (TreeCanonical.locationCount branch n t))
(r : Nat) (ρ : ℝ) (A : ManyGoodRows.RowMap (Block rows upper) r)
(direction : BucketSampler.Direction (rows (h + 1))) (threshold : ℝ) (s : Nat)
abbrev nativeDirection :=
(ProjectedFiberRawPair.direction (CleanDecoderRate.path S) upper (h + 1) d hnode direction).val
def tapeStatistic
(left right : Slots branch (h + 1) → Fin t → MixedSupport.Slot)
(projection : ∀ leaf k, MixedSupport.Projection (left leaf k) (right leaf k))
(tape : WholeCutSampler.Tape rows repeats (CleanDecoderRate.path S)
(CutSlotAssembly.fill (CleanDecoderRate.path S) S.outside right)) : ℝ :=
LowerProjectedOriginalSuccess.kernelSuccess (repeats := repeats) (CleanDecoderRate.path S)
(CutSlotAssembly.fill (CleanDecoderRate.path S) S.outside left)
(CutSlotAssembly.fill (CleanDecoderRate.path S) S.outside right)
(CutProjectionAssembly.fillProjection (CleanDecoderRate.path S) S.outside left right projection)
upper (h + 1) hbranch d hnode direction A S.cut σ
(OriginalDecoderMark.mark F upper (h + 1) κ
(CutSlotAssembly.fill (CleanDecoderRate.path S) S.outside left)) ρ threshold s
(WholeCutSubtree.knownBucketsAtNode rows repeats
(CutSlotAssembly.fill (CleanDecoderRate.path S) S.outside right)
upper (suffix S) (suffix_proper S) (LinearMap.ker A) tape)
(WholeCutSampler.evaluate rows repeats (CleanDecoderRate.path S)
(CutSlotAssembly.fill (CleanDecoderRate.path S) S.outside right) tape)
def childStatistic
(left right : Slots branch (h + 1) → Fin t → MixedSupport.Slot)
(projection : ∀ leaf k, MixedSupport.Projection (left leaf k) (right leaf k))
(children : CutChildGrouping.Raw (C := CleanDecoderRate.Calls S) right rows) : ℝ :=
tapeStatistic S hbranch hnode F κ σ r ρ A direction threshold s left right projection
(CleanPhysicalReplay.physicalTape rows repeats (CleanDecoderRate.path S)
S.outside S.placeholder right S.exterior children)
private theorem childStatistic_congr
(left left' right : Slots branch (h + 1) → Fin t → MixedSupport.Slot)
(hl : left = left')
(projection : ∀ leaf k, MixedSupport.Projection (left leaf k) (right leaf k))
(projection' : ∀ leaf k, MixedSupport.Projection (left' leaf k) (right leaf k))
(hp : ∀ leaf k, HEq (projection leaf k) (projection' leaf k))
(children : CutChildGrouping.Raw (C := CleanDecoderRate.Calls S) right rows) :
childStatistic S hbranch hnode F κ σ r ρ A direction threshold s
left right projection children =
childStatistic S hbranch hnode F κ σ r ρ A direction threshold s
left' right projection' children := by
cases hl
have he : projection = projection' :=
funext (fun leaf => funext (fun k => eq_of_heq (hp leaf k)))
cases he
rfl
theorem rightInput_eq (x : CleanDecoderRate.Sample S) :
CleanPhysicalDecoderLaw.rightInput S (LinearMap.ker A)
(nativeDirection S hnode direction) x =
LowerCutOwnInput.ofArrays (CleanNativeReplay.rightSlots S x) rows upper
(CleanPhysicalDecoderLaw.lower S) (LinearMap.ker A)
(nativeDirection S hnode direction)
(WholeCutSubtree.knownBucketsAtNode rows repeats (CleanNativeReplay.rightSlots S x)
upper (suffix S) (suffix_proper S) (LinearMap.ker A)
(CleanNativeReplay.rightTape S x))
(CleanPhysicalDecoderLaw.rightArrays S x) := by
have hk := WholeCutSubtree.knownBuckets_eq_original rows repeats (Nodes.path upper)
(suffix S) (suffix_proper S) (CleanNativeReplay.rightSlots S x)
(LinearMap.ker A) (CleanNativeReplay.rightTape S x)
exact congrArg
(fun known : HiddenBucketBias.VisibleDirection (LinearMap.ker A) →
OwnInputReference.UpperSpace (CleanNativeReplay.rightSlots S x) upper =>
CleanPhysicalDecoderLaw.inputOfArrays (CleanNativeReplay.rightSlots S x)
upper (CleanPhysicalDecoderLaw.lower S) (LinearMap.ker A)
(nativeDirection S hnode direction) known (CleanPhysicalDecoderLaw.rightArrays S x))
hk.symm
theorem physical_probability_eq (x : CleanDecoderRate.Sample S) :
(CleanPhysicalDecoderLaw.pairKernel S F κ σ r ρ A (LinearMap.ker A)
(nativeDirection S hnode direction) threshold x).probability
(CleanPhysicalHighEvent.highEvent S hbranch s x) =
childStatistic S hbranch hnode F κ σ r ρ A direction threshold s
(CleanPhysicalReplay.leftInside rows repeats (CleanDecoderRate.path S)
S.clauses S.designated x)
(CleanPhysicalReplay.rightInside rows repeats (CleanDecoderRate.path S)
S.clauses S.designated x)
(SourceChildKernel.parentProjection rows S.clauses S.designated
(fun i => ((x i).1, (x i).2.1)) (fun i => (x i).2.2))
(CleanPhysicalReplay.rightBlocks rows repeats (CleanDecoderRate.path S)
S.clauses S.designated x) := by
have ha := CleanPhysicalCanonicalQuery.arrays_pullback rows repeats (CleanDecoderRate.path S)
S.outside S.placeholder S.clauses S.designated x S.exterior
change CleanPhysicalDecoderLaw.leftArrays S x = _ at ha
unfold childStatistic tapeStatistic LowerProjectedOriginalSuccess.kernelSuccess
unfold CleanPhysicalDecoderLaw.pairKernel CleanPhysicalDecoderLaw.leftLaw
CleanPhysicalDecoderLaw.rightLaw
rw [rightInput_eq S hnode r A direction x]
unfold CleanPhysicalDecoderLaw.adviceLaw
rw [ha]
rfl
def sourceStatistic
(sources : SourceChildKernel.Sources (m := m) (t := t) S.designated)
(observed : Σ flags : SourceProjectedTag.Flags (branch := branch) (n := h),
CutChildGrouping.Raw (C := CleanDecoderRate.Calls S)
(SourcePhysicalTaggedChildren.slots S.clauses S.designated sources flags) rows) : ℝ :=
childStatistic S hbranch hnode F κ σ r ρ A direction threshold s
(SourceChildKernel.parentLeftSlots S.clauses S.designated sources)
(SourcePhysicalTaggedChildren.slots S.clauses S.designated sources observed.1)
(fun leaf k => SourceChildKernel.projection S.clauses S.designated leaf.1
(sources leaf.1) (observed.1 leaf.1) leaf.2 k) observed.2
def questionStatistic
(q : PreliminarySampler.Questions branch (h + 1) t m)
(choices : SourceQuestionKernelJoint.ChoiceTuple (branch := branch) (n := h) (t := t))
(flags : SourceProjectedTag.Flags (branch := branch) (n := h))
(children : CutChildGrouping.Raw (C := CleanDecoderRate.Calls S)
(SourceProjectedTag.mixedInside S.clauses S.designated q choices flags) rows) : ℝ :=
childStatistic S hbranch hnode F κ σ r ρ A direction threshold s
(SourceProjectedTag.originalSlots S.clauses q)
(SourceProjectedTag.mixedInside S.clauses S.designated q choices flags)
(SourceProjectedTag.projection S.clauses S.designated q choices flags) children
private theorem sourceStatistic_eq_question
(q : PreliminarySampler.Questions branch (h + 1) t m)
(choices : SourceQuestionKernelJoint.ChoiceTuple (branch := branch) (n := h) (t := t))
(observed : Σ flags : SourceProjectedTag.Flags (branch := branch) (n := h),
CutChildGrouping.Raw (C := CleanDecoderRate.Calls S)
(SourceProjectedTag.mixedInside S.clauses S.designated q choices flags) rows) :
sourceStatistic S hbranch hnode F κ σ r ρ A direction threshold s
(SourceQuestionKernelJoint.sources S.designated q choices) observed =
questionStatistic S hbranch hnode F κ σ r ρ A direction threshold s
q choices observed.1 observed.2 := by
apply childStatistic_congr S hbranch hnode F κ σ r ρ A direction threshold s
_ _ _ (SourceProjectedTag.parentLeftSlots_sources S.clauses S.designated q choices)
intro leaf k
symm
unfold SourceProjectedTag.projection
dsimp only [id]
exact mpr_heq _ _
theorem physical_probability_eq_observed (x : CleanDecoderRate.Sample S) :
(CleanPhysicalDecoderLaw.pairKernel S F κ σ r ρ A (LinearMap.ker A)
(nativeDirection S hnode direction) threshold x).probability
(CleanPhysicalHighEvent.highEvent S hbranch s x) =
let q := SourceQuestionRawSwap.insideQuestions rows S.clauses S.designated x
let choices := SourceQuestionRawSwap.positions rows S.clauses S.designated x
let observed := SourcePhysicalChildExpectation.observe rows S.clauses S.designated q choices
(SourceQuestionRawSwap.raw rows S.clauses S.designated x)
questionStatistic S hbranch hnode F κ σ r ρ A direction threshold s
q choices observed.1 observed.2 := by
let q := SourceQuestionRawSwap.insideQuestions rows S.clauses S.designated x
let choices := SourceQuestionRawSwap.positions rows S.clauses S.designated x
let observed := SourcePhysicalChildExpectation.observe rows S.clauses S.designated q choices
(SourceQuestionRawSwap.raw rows S.clauses S.designated x)
have hread := congrArg
(fun sources : SourceChildKernel.Sources (m := m) (t := t) S.designated =>
sourceStatistic S hbranch hnode F κ σ r ρ A direction threshold s sources
(SourcePhysicalTaggedChildren.observe rows S.clauses S.designated sources
(SourceQuestionRawSwap.raw rows S.clauses S.designated x)))
(SourceQuestionRawSwap.sources_insideQuestions_positions rows S.clauses S.designated x)
exact (physical_probability_eq S hbranch hnode F κ σ r ρ A direction threshold s x).trans
(hread.symm.trans (sourceStatistic_eq_question S hbranch hnode F κ σ r ρ A
direction threshold s q choices observed))
theorem questionStatistic_expectation
(q : PreliminarySampler.Questions branch (h + 1) t m)
(choices : SourceQuestionKernelJoint.ChoiceTuple (branch := branch) (n := h) (t := t))
(flags : SourceProjectedTag.Flags (branch := branch) (n := h)) :
(CutChildGrouping.rawLaw (C := CleanDecoderRate.Calls S)
(SourceProjectedTag.mixedInside S.clauses S.designated q choices flags) rows).expectation
(questionStatistic S hbranch hnode F κ σ r ρ A direction threshold s q choices flags) =
SourceQuestionPhysicalHigh.success S.outside S.placeholder S.clauses S.designated
q choices flags upper (suffix S) (suffix_proper S) (h + 1) hbranch d hnode A S.cut σ
(OriginalDecoderMark.mark F upper (h + 1) κ
(SourceQuestionPhysicalHigh.originalSlots (CleanDecoderRate.path S) S.outside S.clauses q))
ρ threshold S.exterior direction s := by
exact CleanPhysicalRawChildren.expectation_lowerTape rows repeats S.outside S.placeholder
(SourceProjectedTag.mixedInside S.clauses S.designated q choices flags)
upper (suffix S) S.exterior
(tapeStatistic S hbranch hnode F κ σ r ρ A direction threshold s
(SourceProjectedTag.originalSlots S.clauses q)
(SourceProjectedTag.mixedInside S.clauses S.designated q choices flags)
(SourceProjectedTag.projection S.clauses S.designated q choices flags))
theorem mean_probability_eq (cube : Nat) (hcube : 0 < cube) :
(CleanDecoderPairLaw.pairLaw S σ (CleanPhysicalDecoderLaw.useful S F κ)
r ρ A (LinearMap.ker A) (nativeDirection S hnode direction) threshold cube hcube).probability
(fun z => DecoderRankSplit.highAgrees S hbranch s z.1 z.2) =
(PreliminarySampler.questionsLaw
(branch := branch) (n := h + 1) (t := t) (m := m)).expectation
(fun q => (SourceProjectedTag.positionLaw
(branch := branch) (n := h) (t := t)).expectation
(fun choices => (SourceProjectedTag.flagLaw
(fun _ : Fin (branch h) => ProjectedCleanRate.projectionFlag cube hcube)).expectation
(fun flags => SourceQuestionPhysicalHigh.success
S.outside S.placeholder S.clauses S.designated q choices flags
upper (suffix S) (suffix_proper S) (h + 1) hbranch d hnode A S.cut σ
(OriginalDecoderMark.mark F upper (h + 1) κ
(SourceQuestionPhysicalHigh.originalSlots
(CleanDecoderRate.path S) S.outside S.clauses q))
ρ threshold S.exterior direction s))) := by
rw [← CleanPhysicalHighEvent.mean_probability_eq S hbranch F κ σ r ρ A
(LinearMap.ker A) (nativeDirection S hnode direction) threshold cube hcube s]
have hsample := FiniteDistribution.expectation_congr (CleanDecoderRate.rawLaw S cube hcube)
(physical_probability_eq_observed S hbranch hnode F κ σ r ρ A direction threshold s)
refine hsample.trans ?_
rw [show CleanDecoderRate.rawLaw S cube hcube =
SourceChildKernel.originalLaw (C := CleanDecoderRate.Calls S) (t := t)
rows S.clauses S.designated
(fun _ => ProjectedCleanRate.projectionFlag cube hcube) from rfl]
rw [SourcePhysicalChildExpectation.originalLaw_expectation rows S.clauses S.designated
(fun _ => ProjectedCleanRate.projectionFlag cube hcube)
(questionStatistic S hbranch hnode F κ σ r ρ A direction threshold s)]
apply FiniteDistribution.expectation_congr
intro q
apply FiniteDistribution.expectation_congr
intro choices
apply FiniteDistribution.expectation_congr
intro flags
exact questionStatistic_expectation S hbranch hnode F κ σ r ρ A direction threshold s q choices flags
theorem probability_eq_tag_average (cube : Nat) (hcube : 0 < cube) :
(CleanDecoderPairLaw.pairLaw S σ (CleanPhysicalDecoderLaw.useful S F κ)
r ρ A (LinearMap.ker A) (nativeDirection S hnode direction) threshold cube hcube).probability
(fun z => DecoderRankSplit.highAgrees S hbranch s z.1 z.2) =
(PreliminarySampler.questionsLaw
(branch := branch) (n := h + 1) (t := t) (m := m)).expectation
(fun q => (SourceProjectedTag.law (t := t)
(fun _ : Fin (branch h) => ProjectedCleanRate.projectionFlag cube hcube)).expectation
(fun tag => SourceQuestionPhysicalHigh.success
S.outside S.placeholder S.clauses S.designated q tag.1 tag.2
upper (suffix S) (suffix_proper S) (h + 1) hbranch d hnode A S.cut σ
(OriginalDecoderMark.mark F upper (h + 1) κ
(SourceQuestionPhysicalHigh.originalSlots
(CleanDecoderRate.path S) S.outside S.clauses q))
ρ threshold S.exterior direction s)) := by
simpa only [SourceProjectedTag.law, FiniteDistribution.expectation_product] using
mean_probability_eq S hbranch hnode F κ σ r ρ A direction threshold s cube hcube
end
end PerfectCompleteness.CleanPhysicalHighAverage
end
end OAI
|