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import OAI.Computability.PerfectCompleteness.Decoding.FixedStoppedDecoderLaw
import OAI.Computability.PerfectCompleteness.Reduction.HierarchicalFixedAdviceFamilyLemmas
import OAI.Computability.PerfectCompleteness.Sampling.FixedStoppedPhysicalLawLemmas
import OAI.Computability.PerfectCompleteness.Sampling.StoppedProjectedMeetingLaw
namespace OAI
section
namespace PerfectCompleteness.FixedStoppedInverseBound
noncomputable section
open scoped Classical
open RecursiveSpaces TreeSourceSpaces HierarchicalArrays
open UniqueGamesTheorem.Foundations.Games
open UniqueGamesTheorem.Appendix.RankLevelFilter (linearMapFintype)
attribute [local instance] linearMapFintype
variable {δ : ℚ} {hδ : 0 < δ} (parameters : FixedParameters.Parameters δ hδ)
(i j : Fin parameters.plan.depth) (hij : i < j) (input : List Bool)
abbrev clauses := PCPSource.clauseFamily (BinaryLanguage.totalRename input)
abbrev Outer := StoppedProjectedExperiment.Outer
(branch := FixedParameters.branch parameters) (n := parameters.plan.depth)
(j := j.val) (t := FixedRows.sourceLength parameters.plan hδ)
(m := (PCPSource.normalizedFormula (BinaryLanguage.totalRename input)).clauses.length)
abbrev designated := FixedStoppedPhysicalLaw.designated parameters i
abbrev experiment (strategy : FixedPreliminaryGame.Strategy parameters input)
(o : Outer parameters j input) :=
StoppedProjectedExperiment.experiment (clauses input)
(FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij (designated parameters i)
(fun k _ => FixedParameters.branch_pos parameters k)
(StoppedProjectedMeetingLaw.rows_positive parameters.plan)
(FixedStoppedDecoderLaw.flag parameters i)
(FixedStoppedDecoderLaw.labeling parameters input strategy) o
abbrev outerLaw : FiniteDistribution (Outer parameters j input) :=
StoppedProjectedExperiment.outerLaw (Nat.succ_le_of_lt j.isLt)
(fun k _ => FixedParameters.branch_pos parameters k)
abbrev externalLaw (o : Outer parameters j input) :=
StoppedProjectedBuckets.externalLaw (clauses input)
(FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij (designated parameters i)
(StoppedProjectedMeetingLaw.rows_positive parameters.plan) o
(fun k _ => FixedParameters.branch_pos parameters k)
(FixedStoppedDecoderLaw.flag parameters i)
abbrev background (o : Outer parameters j input) :=
StoppedProjectedBuckets.background (clauses input)
(FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij (designated parameters i)
(StoppedProjectedMeetingLaw.rows_positive parameters.plan) o
abbrev scalarLaw (o : Outer parameters j input) :=
StoppedProjectedBuckets.scalarLaw (clauses input)
(FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij (designated parameters i)
(StoppedProjectedMeetingLaw.rows_positive parameters.plan) o
variable (strategy : FixedPreliminaryGame.Strategy parameters input)
local notation "S" => experiment parameters i j hij input strategy
local instance nativeBackgroundFintype (o : Outer parameters j input) :
Fintype (HierarchicalMatrixTable.Background (rows := FixedRows.rows parameters.plan)
(StoppedProjectedExperiment.nativeSlots (clauses input) o)
(StoppedProjectedExperiment.upper (Nat.succ_le_of_lt j.isLt) o)) :=
HierarchicalUsefulCollision.backgroundFintype (rows := FixedRows.rows parameters.plan)
(StoppedProjectedExperiment.nativeSlots (clauses input) o)
(StoppedProjectedExperiment.upper (Nat.succ_le_of_lt j.isLt) o)
local instance nativeRowSpaceFintype (o : Outer parameters j input) :
Fintype (NodeEmbedding.RowSpace
(StoppedProjectedExperiment.nativeSlots (clauses input) o)
(StoppedProjectedExperiment.upper (Nat.succ_le_of_lt j.isLt) o)) :=
HierarchicalUsefulCollision.rowSpaceFintype
(StoppedProjectedExperiment.nativeSlots (clauses input) o)
(StoppedProjectedExperiment.upper (Nat.succ_le_of_lt j.isLt) o)
abbrev PairFiber (o : Outer parameters j input) :=
HierarchicalAgreementMean.PairRecord (rows := FixedRows.rows parameters.plan)
(StoppedProjectedExperiment.nativeSlots (clauses input) o)
(StoppedProjectedExperiment.upper (Nat.succ_le_of_lt j.isLt) o)
abbrev PairSample := (o : Outer parameters j input) × PairFiber parameters j input o
local instance pairFiberFintype (o : Outer parameters j input) :
Fintype (PairFiber parameters j input o) := by
unfold PairFiber HierarchicalAgreementMean.PairRecord
infer_instance
local instance pairSampleFintype : Fintype (PairSample parameters j input) :=
Sigma.instFintype
local instance backgroundFintype (o : Outer parameters j input) :
Fintype (HierarchicalAdviceExperiment.Background (S o)) :=
HierarchicalAdviceExperiment.backgroundFintype (S o)
local instance coarseFintype (o : Outer parameters j input) :
Fintype (HierarchicalAdviceExperiment.Coarse (S o)) :=
HierarchicalAdviceFamily.coarseFintype S o
local instance inputFintype (o : Outer parameters j input)
(bg : HierarchicalAdviceExperiment.Background (S o)) :
Fintype (HierarchicalAdviceExperiment.Input (S o) bg) :=
HierarchicalAdviceExperiment.inputFintype (S o) bg
local instance inputFiniteDimensional (o : Outer parameters j input)
(bg : HierarchicalAdviceExperiment.Background (S o)) :
FiniteDimensional F2 (HierarchicalAdviceExperiment.Input (S o) bg) :=
HierarchicalAdviceExperiment.inputFiniteDimensional (S o) bg
local instance projectedSampleFintype (o : Outer parameters j input) :
Fintype (HierarchicalProjectedExperiment.Sample
(rows := FixedRows.rows parameters.plan) (repeats := FixedRows.repeats parameters.plan)
(StoppedProjectedExperiment.projectedSlots (clauses input) (FixedRows.rows parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij (designated parameters i) o)
(StoppedProjectedExperiment.upper (Nat.succ_le_of_lt j.isLt) o)
(StoppedProjectedExperiment.lower (FixedRows.rows parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij o)
(StoppedProjectedExperiment.cut (FixedRows.rows parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij o)) := by
letI : (k : StoppedProjectedExperiment.Inner
(branch := FixedParameters.branch parameters) (n := parameters.plan.depth)
(i := i.val) (j := j.val) (t := FixedRows.sourceLength parameters.plan hδ)
(FixedRows.rows parameters.plan)) →
Fintype (WholeArraySampler.Tape (FixedRows.rows parameters.plan)
(FixedRows.repeats parameters.plan)
(WholeArrayInteriorOwnInputLaw.fullPath
(StoppedProjectedExperiment.upper (Nat.succ_le_of_lt j.isLt) o)
(StoppedProjectedExperiment.lower (FixedRows.rows parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij o k)
(StoppedProjectedExperiment.cut (FixedRows.rows parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij o k))
(StoppedProjectedExperiment.projectedSlots (clauses input) (FixedRows.rows parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij (designated parameters i) o k)) :=
fun k => WholeArraySampler.tapeFintype _ _ _ _
exact Sigma.instFintype
local instance adviceFintype (o : Outer parameters j input) (r : Nat) :
Fintype (HierarchicalAdviceExperiment.Advice (S o) r) := linearMapFintype
local instance originalFiberFintype (o : Outer parameters j input) (r : Nat) :
Fintype (HierarchicalProjectedExperiment.Sample
(rows := FixedRows.rows parameters.plan) (repeats := FixedRows.repeats parameters.plan)
(StoppedProjectedExperiment.projectedSlots (clauses input) (FixedRows.rows parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij (designated parameters i) o)
(StoppedProjectedExperiment.upper (Nat.succ_le_of_lt j.isLt) o)
(StoppedProjectedExperiment.lower (FixedRows.rows parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij o)
(StoppedProjectedExperiment.cut (FixedRows.rows parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij o) ×
HierarchicalAdviceExperiment.Advice (S o) r) := by
infer_instance
abbrev backgrounds :=
HierarchicalBucketFamily.actualBackgrounds (experiment parameters i j hij input strategy)
(externalLaw parameters i j hij input) (background parameters i j hij input)
abbrev usefulMass : ℝ :=
HierarchicalBucketFamily.usefulMass (experiment parameters i j hij input strategy)
(InitialParameters.useful δ) (outerLaw parameters j input)
(externalLaw parameters i j hij input) (background parameters i j hij input)
(scalarLaw parameters i j hij input)
(HierarchicalFixedAdviceFamily.rows_positive parameters
(experiment parameters i j hij input strategy))
abbrev actualPairs : FiniteDistribution (PairSample parameters j input) :=
HierarchicalBucketFamily.actualPairLaw (experiment parameters i j hij input strategy)
(outerLaw parameters j input) (externalLaw parameters i j hij input)
(background parameters i j hij input) (scalarLaw parameters i j hij input)
(HierarchicalFixedAdviceFamily.rows_positive parameters
(experiment parameters i j hij input strategy))
abbrev referencePairs : FiniteDistribution (PairSample parameters j input) :=
HierarchicalBucketFamily.referencePairLaw (experiment parameters i j hij input strategy)
(outerLaw parameters j input) (backgrounds parameters i j hij input strategy)
(HierarchicalFixedAdviceFamily.rows_positive parameters
(experiment parameters i j hij input strategy))
abbrev pairedVariation : ℝ :=
(actualPairs parameters i j hij input strategy).totalVariation
(referencePairs parameters i j hij input strategy)
abbrev coarseVariation : ℝ :=
((HierarchicalAdviceFamily.originalLaw (experiment parameters i j hij input strategy)
(outerLaw parameters j input) parameters.plan.order).pushforward
(HierarchicalAdviceFamily.observe (experiment parameters i j hij input strategy)
parameters.plan.order)).totalVariation
((HierarchicalAdviceFamily.referenceLaw (experiment parameters i j hij input strategy)
(outerLaw parameters j input) (backgrounds parameters i j hij input strategy)
parameters.plan.order).pushforward
(HierarchicalAdviceFamily.referenceObserve
(experiment parameters i j hij input strategy) parameters.plan.order))
abbrev expectedMeeting : ℝ :=
StoppedProjectedMeetingLaw.expectedMeeting parameters.plan (clauses input)
(Nat.succ_le_of_lt j.isLt) hij (designated parameters i)
(fun k _ => FixedParameters.branch_pos parameters k)
(FixedStoppedDecoderLaw.flag parameters i)
(FixedStoppedDecoderLaw.labeling parameters input strategy)
theorem expectedMeeting_ge_gamma
(hmass : InitialParameters.simultaneous δ / 2 ≤
usefulMass parameters i j hij input strategy)
(hpaired : pairedVariation parameters i j hij input strategy ≤ 10 * parameters.accuracy)
(hcoarse : coarseVariation parameters i j hij input strategy ≤ 10 * parameters.accuracy) :
FixedRankContradiction.gamma parameters (j.val + 1) ≤
expectedMeeting parameters i j hij input strategy := by
have hheight (o : Outer parameters j input) :
Nodes.height (experiment parameters i j hij input strategy o).upper = j.val + 1 :=
StoppedProjectedExperiment.upper_height (Nat.succ_le_of_lt j.isLt) o
have hbound := HierarchicalFixedAdviceFamily.prediction_difference parameters
(experiment parameters i j hij input strategy) (outerLaw parameters j input)
(externalLaw parameters i j hij input) (background parameters i j hij input)
(scalarLaw parameters i j hij input) (backgrounds parameters i j hij input strategy)
(j.val + 1) (Nat.succ_le_of_lt j.isLt) hheight hmass hpaired hcoarse
have hmeeting := HierarchicalProjectedOuterAverage.expectedMeeting_ge_prediction_difference
parameters.plan (StoppedProjectedExperiment.nativeSlots (clauses input))
(StoppedProjectedExperiment.projectedSlots (clauses input) (FixedRows.rows parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij (designated parameters i))
(StoppedProjectedExperiment.projection (clauses input) (FixedRows.rows parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij (designated parameters i))
(StoppedProjectedExperiment.upper (Nat.succ_le_of_lt j.isLt)) (i.val + 1)
(StoppedProjectedExperiment.lower (FixedRows.rows parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij)
(StoppedProjectedExperiment.cut (FixedRows.rows parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij)
(StoppedProjectedExperiment.direction (FixedRows.rows parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij)
(fun _ => StoppedProjectedExperiment.innerLaw (FixedRows.rows parameters.plan) hij
(fun k _ => FixedParameters.branch_pos parameters k)
(StoppedProjectedMeetingLaw.rows_positive parameters.plan)
(FixedStoppedDecoderLaw.flag parameters i))
(outerLaw parameters j input) (FixedStoppedDecoderLaw.labeling parameters input strategy)
hδ (FixedRows.rows parameters.plan (j.val + 1))
(fun o => congrArg (FixedRows.rows parameters.plan)
(StoppedProjectedExperiment.upper_height (Nat.succ_le_of_lt j.isLt) o))
calc
_ = HierarchicalProjectedOuterAverage.factor parameters.plan
(FixedRows.rows parameters.plan (j.val + 1)) *
UpperParameterScalars.b (InitialParameters.useful δ) parameters.plan.density
(InitialParameters.inverse δ) parameters.plan.order
(FixedRows.rows parameters.plan (j.val + 1)) :=
(HierarchicalProjectedOuterAverage.factor_mul_b_eq_gamma parameters.plan _).symm
_ ≤ HierarchicalProjectedOuterAverage.factor parameters.plan
(FixedRows.rows parameters.plan (j.val + 1)) *
HierarchicalFixedAdviceFamily.predictionDifference parameters
(experiment parameters i j hij input strategy) (outerLaw parameters j input) :=
mul_le_mul_of_nonneg_left hbound
(HierarchicalProjectedOuterAverage.factor_nonneg parameters.plan _)
_ ≤ _ := hmeeting
end
end PerfectCompleteness.FixedStoppedInverseBound
end
end OAI
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