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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 | import OAI.Computability.PerfectCompleteness.Decoding.FixedProjectionErrorLemmas
import OAI.Computability.PerfectCompleteness.Sampling.SourceChildMarkedLaw
namespace OAI
section
namespace PerfectCompleteness.FixedSourceChildCollision
noncomputable section
open scoped Classical BigOperators
open FixedParameters FixedRows RecursiveSpaces DescendantSpaces TreeSourceSpaces
open UniqueGamesTheorem.Foundations.Games
variable {δ : ℚ} {hδ : 0 < δ} (parameters : Parameters δ hδ)
{n height v m : Nat} [NeZero m]
(path : Path (branch parameters) n (height + 1))
(clauses : Fin m → SourceClause.NormalizedClause v)
(designated : Fin (branch parameters height) → Slots (branch parameters) height)
abbrev Calls := Option (WholeCutCalls.Index (rows parameters.plan) (repeats parameters.plan) path)
abbrev Questions := SourceChildMarkedLaw.Questions
(branch := branch parameters) (n := height)
(t := sourceLength parameters.plan hδ) (m := m)
abbrev Native (q : Questions (m := m) parameters (height := height)) :=
CutChildGrouping.Raw (C := Calls parameters path)
(SourceChildMarkedLaw.questionSlots clauses q) (rows parameters.plan)
abbrev Form (q : Questions (m := m) parameters (height := height)) :=
ChildBilinearCollisionTransfer.ParentForm (SourceChildMarkedLaw.questionSlots clauses q)
abbrev Sample := SourceChildKernel.Sample (C := Calls parameters path)
(t := sourceLength parameters.plan hδ) (rows parameters.plan) clauses designated
def markedLaw : FiniteDistribution (Sample parameters path clauses designated) :=
SourceChildKernel.originalLaw (C := Calls parameters path)
(t := sourceLength parameters.plan hδ) (rows parameters.plan) clauses designated
(fun _ => ProjectionPosterior.bernoulli (projectionProbability parameters height : ℝ)
(by exact_mod_cast (projectionProbability_bounds parameters height).1.le)
(by exact_mod_cast (projectionProbability_bounds parameters height).2.le))
def uniformLaw : FiniteDistribution
(Σ q : Questions (m := m) parameters (height := height), Native parameters path clauses q) :=
CompletionSoundness.sigmaLaw
(PreliminarySampler.questionsLaw (branch := branch parameters) (n := height + 1)
(t := sourceLength parameters.plan hδ) (m := m))
(fun q => CutChildGrouping.rawLaw (C := Calls parameters path)
(SourceChildMarkedLaw.questionSlots clauses q) (rows parameters.plan))
variable (F₀ F₁ : (q : Questions (m := m) parameters (height := height)) →
Native parameters path clauses q → Option (Form parameters clauses q))
def restrictedEvent (sample : Sample parameters path clauses designated) : Bool :=
let record := SourceChildMarkedLaw.observe (rows parameters.plan) clauses designated sample
BilinearCollisionTransfer.restrictedCollision
(F₀ record.1 record.2.2) (F₁ record.1 record.2.2)
(ChildUnmarkedSpace.space (SourceChildMarkedLaw.questionSlots clauses record.1) record.2.1)
def fullEvent (record :
Σ q : Questions (m := m) parameters (height := height), Native parameters path clauses q) : Bool :=
BilinearCollisionTransfer.fullCollision (F₀ record.1 record.2) (F₁ record.1 record.2)
private theorem expectation_add_const {A : Type*} [Fintype A]
(μ : FiniteDistribution A) (f : A → 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 restricted_probability_le (hn : n ≤ parameters.plan.depth) :
(markedLaw parameters path clauses designated).probability
(restrictedEvent parameters path clauses designated F₀ F₁) ≤
(uniformLaw parameters path clauses).probability
(fullEvent parameters path clauses F₀ F₁) + 2 * parameters.accuracy := by
have hβ : 0 ≤ (projectionProbability parameters height : ℝ) := by
exact_mod_cast (projectionProbability_bounds parameters height).1.le
have hβ' : (projectionProbability parameters height : ℝ) ≤ 1 := by
exact_mod_cast (projectionProbability_bounds parameters height).2.le
have hlaw := SourceChildMarkedLaw.original_extended_marked_law
(t := sourceLength parameters.plan hδ) (rows parameters.plan) clauses designated
(repeats parameters.plan) path (projectionProbability parameters height : ℝ) hβ hβ'
have hprob := congrArg
(fun μ : FiniteDistribution
(SourceChildMarkedLaw.Observation (C := Calls parameters path)
(t := sourceLength parameters.plan hδ) (rows parameters.plan) clauses) =>
μ.probability (fun record => BilinearCollisionTransfer.restrictedCollision
(F₀ record.1 record.2.2) (F₁ record.1 record.2.2)
(ChildUnmarkedSpace.space (SourceChildMarkedLaw.questionSlots clauses record.1)
record.2.1))) hlaw
rw [FiniteDistribution.probability_pushforward,
CompletionSoundness.sigmaLaw_probability] at hprob
rw [show (markedLaw parameters path clauses designated).probability
(restrictedEvent parameters path clauses designated F₀ F₁) = _ from hprob]
rw [uniformLaw, CompletionSoundness.sigmaLaw_probability]
have hpoint := fun q : Questions (m := m) parameters (height := height) =>
(FixedChildCollisionTransfer.marked_restricted_lt_uniform_full parameters
(FixedChildCollisionTransfer.lower_pair_calls_le parameters path hn)
(SourceChildMarkedLaw.questionSlots clauses q)
(SourceChildMarkedLaw.projectedSlots clauses designated q)
(SourceChildMarkedLaw.questionProjection clauses designated q)
(fun _ => FiniteDistribution.uniform
(SourceChildMarkedLaw.Choice (branch := branch parameters) (n := height)
(t := sourceLength parameters.plan hδ)))
(F₀ q) (F₁ q)).le
have havg := SmallBias.expectation_mono
(PreliminarySampler.questionsLaw (branch := branch parameters) (n := height + 1)
(t := sourceLength parameters.plan hδ) (m := m)) hpoint
rw [expectation_add_const] at havg
simpa only [← SourceChildMarkedLaw.replacementLaws_eq, fullEvent] using havg
end
end PerfectCompleteness.FixedSourceChildCollision
end
end OAI
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