File size: 5,130 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 | import OAI.Computability.PerfectCompleteness.Construction.HiddenBucketBias
import OAI.Computability.PerfectCompleteness.Reduction.PreliminaryAveraging
import OAI.Computability.PerfectCompleteness.Sampling.StoppedUsefulPairLaw
namespace OAI
section
namespace PerfectCompleteness.StoppedBucketUsefulProbability
noncomputable section
open scoped Classical
open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays
open OriginalWholeCutTape WholeArrayInteriorExterior
open UniqueGamesTheorem.Foundations.Games
open UniqueGamesTheorem.Appendix.RankLevelFilter (linearMapFintype)
attribute [local instance] linearMapFintype
private theorem probability_product_fst {A B : Type*} [Fintype A] [Fintype B]
(μ : FiniteDistribution A) (ν : FiniteDistribution B) (event : A → Bool) :
(μ.product ν).probability (fun z => event z.1) = μ.probability event := by
have h := FiniteDistribution.probability_pushforward (μ.product ν) Prod.fst event
rw [HiddenBucketBias.product_fst] at h
exact h.symm
variable {branch : Nat → Nat} {N j i t : Nat}
(rows repeats : Nat → Nat) (p : Path branch N (j + 1))
(chosen : Fin (branch j)) (q : Path branch j i)
(native right : Slots branch N → Fin t → MixedSupport.Slot)
(projection : ∀ s a, MixedSupport.Projection (native s a) (right s a))
local instance rowSpaceFintype : Fintype (NodeEmbedding.RowSpace native (upperNode p)) :=
Fintype.ofFinite _
local instance backgroundFintype :
Fintype (HierarchicalMatrixTable.Background (rows := rows) native (upperNode p)) :=
Fintype.ofFinite _
variable (hrows : 0 < rows (j + 1)) (lowerLevel : Nat) {Ω : Type*} [Fintype Ω]
(original : FiniteDistribution Ω) (arrays : Ω → Arrays native rows)
(lowerEvent : Ω → Bool) (κ : ℝ)
(σ : KeyStrategy.Strategy (TreeCanonical.locationCount branch N t))
theorem usefulEvent_observe
(sample : StoppedUsefulPairLaw.Sample rows repeats p chosen q right) :
HierarchicalUsefulCollision.usefulEvent native (upperNode p) lowerLevel
original arrays lowerEvent κ σ
(StoppedUsefulPairLaw.background rows repeats p chosen q native right projection)
(StoppedUsefulPairLaw.observe rows repeats p chosen q native right projection sample) =
HierarchicalUsefulCollision.usefulAccepted native (upperNode p) lowerLevel
original arrays lowerEvent κ σ
(StoppedUsefulPairLaw.background rows repeats p chosen q native right projection
(StoppedUsefulPairLaw.readout rows repeats p chosen q native right projection sample.2.1).1)
(CutNodeRows.directionEquiv rows p sample.1)
(CutNodeRows.bucketTapeEquiv rows p native
(StoppedUsefulPairLaw.readout rows repeats p chosen q native right projection sample.2.1).2) :=
rfl
theorem useful_probability_eq :
(HierarchicalUsefulCollision.law native (upperNode p)
(StoppedUsefulPairLaw.externalLaw rows repeats p chosen q right)
(StoppedUsefulPairLaw.scalarLaw rows repeats p chosen q native right projection)
(by simpa only [upperNode_height] using hrows)).probability
(HierarchicalUsefulCollision.usefulEvent native (upperNode p) lowerLevel
original arrays lowerEvent κ σ
(StoppedUsefulPairLaw.background rows repeats p chosen q native right projection)) =
(StoppedUsefulPairLaw.directionLaw rows j hrows).expectation (fun a =>
(WholeArraySampler.tapeLaw rows repeats (p.append (.step chosen q)) right).probability
(fun tape => HierarchicalUsefulCollision.usefulAccepted native (upperNode p) lowerLevel
original arrays lowerEvent κ σ
(StoppedUsefulPairLaw.background rows repeats p chosen q native right projection
(StoppedUsefulPairLaw.readout rows repeats p chosen q native right projection tape).1)
(CutNodeRows.directionEquiv rows p a)
(CutNodeRows.bucketTapeEquiv rows p native
(StoppedUsefulPairLaw.readout rows repeats p chosen q native right projection tape).2))) := by
rw [← StoppedUsefulPairLaw.observe_law rows repeats p chosen q native right projection hrows,
FiniteDistribution.probability_pushforward]
simp_rw [usefulEvent_observe]
rw [StoppedUsefulPairLaw.actualLaw, PreliminaryAveraging.probability_product]
apply FiniteDistribution.expectation_congr
intro a
exact probability_product_fst
(WholeArraySampler.tapeLaw rows repeats (p.append (.step chosen q)) right)
(RecursiveSampler.law F2 repeats (.step chosen q) (LeafDomain (cutSlots p right)))
(fun tape => HierarchicalUsefulCollision.usefulAccepted native (upperNode p) lowerLevel
original arrays lowerEvent κ σ
(StoppedUsefulPairLaw.background rows repeats p chosen q native right projection
(StoppedUsefulPairLaw.readout rows repeats p chosen q native right projection tape).1)
(CutNodeRows.directionEquiv rows p a)
(CutNodeRows.bucketTapeEquiv rows p native
(StoppedUsefulPairLaw.readout rows repeats p chosen q native right projection tape).2))
end
end PerfectCompleteness.StoppedBucketUsefulProbability
end
end OAI
|