openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Sampling /FixedSourcePairVariation.lean
| import OAI.Computability.PerfectCompleteness.Construction.HierarchicalBucketFamily | |
| import OAI.Computability.PerfectCompleteness.Decoding.FixedUpperPairErrorLemmas | |
| import OAI.Computability.PerfectCompleteness.Foundations.HierarchicalUsefulBackground | |
| import OAI.Computability.PerfectCompleteness.Sampling.CommonProductVariationLemmas | |
| import OAI.Computability.PerfectCompleteness.Sampling.StoppedSourcePairLaw | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.FixedSourcePairVariation | |
| open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays | |
| open OriginalWholeCutTape WholeArrayInteriorExterior | |
| open UniqueGamesTheorem.Foundations.Games | |
| open UniqueGamesTheorem.Appendix.RankLevelFilter (linearMapFintype) | |
| open scoped Classical | |
| noncomputable section | |
| attribute [local instance] linearMapFintype | |
| attribute [local instance 2000] HierarchicalUsefulCollision.backgroundFintype | |
| HierarchicalUsefulCollision.rowSpaceFintype | |
| private theorem sigma_variation_le_const | |
| {O : Type*} [Fintype O] {R : O → Type*} [∀ o, Fintype (R o)] | |
| (outer : FiniteDistribution O) | |
| (P Q : (o : O) → FiniteDistribution (R o)) | |
| (error : ℝ) (bound : ∀ o, (P o).totalVariation (Q o) ≤ error) : | |
| (CompletionSoundness.sigmaLaw outer P).totalVariation | |
| (CompletionSoundness.sigmaLaw outer Q) ≤ error := by | |
| rw [ConditionalVariation.sigma_totalVariation outer P Q] | |
| calc | |
| outer.expectation (fun o => (P o).totalVariation (Q o)) ≤ | |
| outer.expectation (fun _ => error) := | |
| SmallBias.expectation_mono outer bound | |
| _ = error := SmallBias.expectation_const outer error | |
| variable {δ : ℚ} {hδ : 0 < δ} (parameters : FixedParameters.Parameters δ hδ) | |
| {n i j v m : Nat} | |
| abbrev Outer := StoppedProjectedExperiment.Outer | |
| (branch := FixedParameters.branch parameters) (n := n) (j := j) | |
| (t := FixedRows.sourceLength parameters.plan hδ) (m := m) | |
| def flags : Fin (FixedParameters.branch parameters i) → FiniteDistribution Bool := | |
| fun _ => ProjectionPosterior.bernoulli (FixedParameters.projectionProbability parameters i : ℝ) | |
| (by exact_mod_cast (FixedParameters.projectionProbability_bounds parameters i).1.le) | |
| (by exact_mod_cast (FixedParameters.projectionProbability_bounds parameters i).2.le) | |
| abbrev positiveRows (height : Nat) : 0 < FixedRows.rows parameters.plan (height + 1) := | |
| FixedUpperCutTriangle.rows_positive parameters (height + 1) | |
| variable (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (hupper : j + 1 ≤ n) (hij : i < j) | |
| (designated : Fin (FixedParameters.branch parameters i) → Slots (FixedParameters.branch parameters) i) | |
| abbrev experiment | |
| (σ : KeyStrategy.Strategy (TreeCanonical.locationCount (FixedParameters.branch parameters) n | |
| (FixedRows.sourceLength parameters.plan hδ))) | |
| (o : Outer (n := n) (j := j) (m := m) parameters) := | |
| StoppedProjectedBuckets.experiment clauses (FixedRows.rows parameters.plan) | |
| (FixedRows.repeats parameters.plan) hupper hij designated (positiveRows parameters) o | |
| (fun k _ => FixedParameters.branch_pos parameters k) (flags (i := i) parameters) σ | |
| abbrev Descriptor (o : Outer (n := n) (j := j) (m := m) parameters) := | |
| StoppedProjectedBuckets.Descriptor clauses (FixedRows.rows parameters.plan) | |
| (FixedRows.repeats parameters.plan) hupper hij designated (positiveRows parameters) o | |
| def externalLaw (o : Outer (n := n) (j := j) (m := m) parameters) : | |
| FiniteDistribution (Descriptor parameters clauses hupper hij designated o) := | |
| StoppedProjectedBuckets.externalLaw clauses (FixedRows.rows parameters.plan) | |
| (FixedRows.repeats parameters.plan) hupper hij designated (positiveRows parameters) o | |
| (fun k _ => FixedParameters.branch_pos parameters k) (flags (i := i) parameters) | |
| abbrev background (o : Outer (n := n) (j := j) (m := m) parameters) := | |
| StoppedProjectedBuckets.background clauses (FixedRows.rows parameters.plan) | |
| (FixedRows.repeats parameters.plan) hupper hij designated (positiveRows parameters) o | |
| abbrev scalarLaw (o : Outer (n := n) (j := j) (m := m) parameters) := | |
| StoppedProjectedBuckets.scalarLaw clauses (FixedRows.rows parameters.plan) | |
| (FixedRows.repeats parameters.plan) hupper hij designated (positiveRows parameters) o | |
| abbrev upperRows (o : Outer (n := n) (j := j) (m := m) parameters) := | |
| StoppedProjectedBuckets.rows_positive (FixedRows.rows parameters.plan) hupper (positiveRows parameters) o | |
| variable (σ : KeyStrategy.Strategy (TreeCanonical.locationCount (FixedParameters.branch parameters) n | |
| (FixedRows.sourceLength parameters.plan hδ))) | |
| local instance backgroundFintype (o : Outer (n := n) (j := j) (m := m) parameters) : | |
| Fintype (HierarchicalAgreementMean.Background (rows := FixedRows.rows parameters.plan) | |
| (StoppedProjectedExperiment.nativeSlots clauses o) | |
| (StoppedProjectedExperiment.upper hupper o)) := Fintype.ofFinite _ | |
| local instance rowSpaceFintype (o : Outer (n := n) (j := j) (m := m) parameters) : | |
| Fintype (NodeEmbedding.RowSpace (StoppedProjectedExperiment.nativeSlots clauses o) | |
| (StoppedProjectedExperiment.upper hupper o)) := Fintype.ofFinite _ | |
| abbrev PairFiber (o : Outer (n := n) (j := j) (m := m) parameters) := | |
| HierarchicalAgreementMean.PairRecord (rows := FixedRows.rows parameters.plan) | |
| (StoppedProjectedExperiment.nativeSlots clauses o) | |
| (StoppedProjectedExperiment.upper hupper o) | |
| abbrev PairRecord := | |
| (o : Outer (n := n) (j := j) (m := m) parameters) × PairFiber parameters clauses hupper o | |
| theorem fiber_lt_six (hroot : n ≤ parameters.plan.depth) | |
| (o : Outer (n := n) (j := j) (m := m) parameters) : | |
| (HierarchicalAdviceFromBuckets.pairedLaw (experiment parameters clauses hupper hij designated σ o) | |
| (externalLaw parameters clauses hupper hij designated o) | |
| (background parameters clauses hupper hij designated o) | |
| (scalarLaw parameters clauses hupper hij designated o) (upperRows parameters hupper o)).totalVariation | |
| (HierarchicalAgreementMean.referenceLaw | |
| (StoppedProjectedExperiment.nativeSlots clauses o) (StoppedProjectedExperiment.upper hupper o) | |
| ((externalLaw parameters clauses hupper hij designated o).pushforward | |
| (background parameters clauses hupper hij designated o)) (upperRows parameters hupper o)) < | |
| 6 * parameters.accuracy := by | |
| let pref := StoppedProjectedExperiment.upperPath hupper o | |
| let slots := StoppedProjectedExperiment.nativeSlots clauses o | |
| let projected := SourceProjectedWholeChoices.projected clauses hupper hij o | |
| let projection := SourceProjectedWholeChoices.projection clauses hupper hij o | |
| let choiceLaw := fun _ : UpperCutTriangle.Prefix (FixedParameters.branch parameters) j i => | |
| SourceProjectedChoices.choiceLaw | |
| (branch := FixedParameters.branch parameters) (n := i) | |
| (t := FixedRows.sourceLength parameters.plan hδ) | |
| have hpair := StoppedSourcePairLaw.pairedLaw_eq_modified clauses | |
| (FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan) hupper hij designated | |
| (positiveRows parameters) o (fun k _ => FixedParameters.branch_pos parameters k) | |
| (FixedParameters.projectionProbability parameters i : ℝ) | |
| (by exact_mod_cast (FixedParameters.projectionProbability_bounds parameters i).1.le) | |
| (by exact_mod_cast (FixedParameters.projectionProbability_bounds parameters i).2.le) σ | |
| change HierarchicalAdviceFromBuckets.pairedLaw | |
| (experiment parameters clauses hupper hij designated σ o) | |
| (externalLaw parameters clauses hupper hij designated o) | |
| (background parameters clauses hupper hij designated o) | |
| (scalarLaw parameters clauses hupper hij designated o) (upperRows parameters hupper o) = | |
| FixedUpperCutTriangle.modifiedPairLaw parameters pref (Nat.succ_le_succ hij.le) slots | |
| projected projection choiceLaw at hpair | |
| have h := FixedUpperCutTriangle.original_reference_lt parameters pref | |
| (Nat.succ_le_succ hij.le) slots projected projection choiceLaw hroot | |
| change (FixedUpperCutTriangle.modifiedPairLaw parameters pref (Nat.succ_le_succ hij.le) slots | |
| projected projection choiceLaw).totalVariation | |
| (HierarchicalAgreementMean.referenceLaw slots (upperNode pref) | |
| ((FixedUpperCutTriangle.modifiedPairLaw parameters pref (Nat.succ_le_succ hij.le) slots | |
| projected projection choiceLaw).pushforward Prod.fst) (upperRows parameters hupper o)) < | |
| 6 * parameters.accuracy at h | |
| rw [← hpair] at h | |
| have hbackground : | |
| (HierarchicalAdviceFromBuckets.pairedLaw | |
| (experiment parameters clauses hupper hij designated σ o) | |
| (externalLaw parameters clauses hupper hij designated o) | |
| (background parameters clauses hupper hij designated o) | |
| (scalarLaw parameters clauses hupper hij designated o) | |
| (upperRows parameters hupper o)).pushforward Prod.fst = | |
| (externalLaw parameters clauses hupper hij designated o).pushforward | |
| (background parameters clauses hupper hij designated o) := | |
| HierarchicalUsefulBackground.pairLaw_background slots (upperNode pref) | |
| (externalLaw parameters clauses hupper hij designated o) | |
| (background parameters clauses hupper hij designated o) | |
| (scalarLaw parameters clauses hupper hij designated o) (upperRows parameters hupper o) | |
| have hvariation := congrArg | |
| (fun backgroundLaw : FiniteDistribution | |
| (HierarchicalAgreementMean.Background (rows := FixedRows.rows parameters.plan) | |
| slots (upperNode pref)) => | |
| (HierarchicalUsefulCollision.pairLaw slots (upperNode pref) | |
| (externalLaw parameters clauses hupper hij designated o) | |
| (background parameters clauses hupper hij designated o) | |
| (scalarLaw parameters clauses hupper hij designated o) | |
| (upperRows parameters hupper o)).totalVariation | |
| (HierarchicalAgreementMean.referenceLaw slots (upperNode pref) | |
| backgroundLaw (upperRows parameters hupper o))) hbackground | |
| exact lt_of_eq_of_lt hvariation.symm h | |
| def actualLaw (outer : FiniteDistribution (Outer (n := n) (j := j) (m := m) parameters)) : | |
| FiniteDistribution (PairRecord parameters clauses hupper) := | |
| HierarchicalBucketFamily.actualPairLaw (experiment parameters clauses hupper hij designated σ) outer | |
| (externalLaw parameters clauses hupper hij designated) | |
| (background parameters clauses hupper hij designated) | |
| (scalarLaw parameters clauses hupper hij designated) (upperRows parameters hupper) | |
| def referenceLaw (outer : FiniteDistribution (Outer (n := n) (j := j) (m := m) parameters)) : | |
| FiniteDistribution (PairRecord parameters clauses hupper) := | |
| HierarchicalBucketFamily.referencePairLaw (experiment parameters clauses hupper hij designated σ) outer | |
| (HierarchicalBucketFamily.actualBackgrounds (experiment parameters clauses hupper hij designated σ) | |
| (externalLaw parameters clauses hupper hij designated) | |
| (background parameters clauses hupper hij designated)) (upperRows parameters hupper) | |
| theorem totalVariation_le_six (hroot : n ≤ parameters.plan.depth) | |
| (outer : FiniteDistribution (Outer (n := n) (j := j) (m := m) parameters)) : | |
| (actualLaw parameters clauses hupper hij designated σ outer).totalVariation | |
| (referenceLaw parameters clauses hupper hij designated σ outer) ≤ 6 * parameters.accuracy := by | |
| exact sigma_variation_le_const | |
| (R := PairFiber parameters clauses hupper) outer | |
| (fun o => HierarchicalAdviceFromBuckets.pairedLaw | |
| (experiment parameters clauses hupper hij designated σ o) | |
| (externalLaw parameters clauses hupper hij designated o) | |
| (background parameters clauses hupper hij designated o) | |
| (scalarLaw parameters clauses hupper hij designated o) (upperRows parameters hupper o)) | |
| (fun o => HierarchicalAgreementMean.referenceLaw | |
| (StoppedProjectedExperiment.nativeSlots clauses o) (StoppedProjectedExperiment.upper hupper o) | |
| ((externalLaw parameters clauses hupper hij designated o).pushforward | |
| (background parameters clauses hupper hij designated o)) (upperRows parameters hupper o)) | |
| (6 * parameters.accuracy) | |
| (fun o => (fiber_lt_six parameters clauses hupper hij designated σ hroot o).le) | |
| theorem totalVariation_le_ten (hroot : n ≤ parameters.plan.depth) | |
| (outer : FiniteDistribution (Outer (n := n) (j := j) (m := m) parameters)) : | |
| (actualLaw parameters clauses hupper hij designated σ outer).totalVariation | |
| (referenceLaw parameters clauses hupper hij designated σ outer) ≤ 10 * parameters.accuracy := by | |
| have h := totalVariation_le_six parameters clauses hupper hij designated σ hroot outer | |
| have hpositive := parameters.accuracy_pos | |
| linarith | |
| end | |
| end PerfectCompleteness.FixedSourcePairVariation | |
| end | |
| end OAI | |