openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Construction /StoppedProjectedBuckets.lean
| import OAI.Computability.PerfectCompleteness.Construction.HierarchicalAdviceFromBuckets | |
| import OAI.Computability.PerfectCompleteness.Construction.StoppedProjectedBucketIndex | |
| import OAI.Computability.PerfectCompleteness.Foundations.StoppedPathHead | |
| import OAI.Computability.PerfectCompleteness.Sampling.StoppedUsefulPairLaw | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.StoppedProjectedBuckets | |
| noncomputable section | |
| open scoped Classical | |
| open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays | |
| open UniqueGamesTheorem.Foundations.Games | |
| open UniqueGamesTheorem.Appendix.RankLevelFilter (linearMapFintype) | |
| attribute [local instance] linearMapFintype | |
| variable {branch : Nat → Nat} {n i j t v m : Nat} | |
| (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (rows repeats : Nat → Nat) (hupper : j + 1 ≤ n) (hij : i < j) | |
| (designated : Fin (branch i) → Slots branch i) | |
| (hrows : ∀ k, 0 < rows (k + 1)) | |
| (o : StoppedProjectedExperiment.Outer | |
| (branch := branch) (n := n) (j := j) (t := t) (m := m)) | |
| abbrev Base := StoppedProjectedBucketIndex.Base (branch := branch) (i := i) (j := j) (t := t) | |
| abbrev key (base : Base (branch := branch) (i := i) (j := j) (t := t)) := | |
| StoppedProjectedBucketIndex.key (n := n) rows hrows base | |
| abbrev lowerPath (base : Base (branch := branch) (i := i) (j := j) (t := t)) := | |
| StoppedProjectedExperiment.lowerPath rows hij (key (n := n) rows hrows base) | |
| def chosen (base : Base (branch := branch) (i := i) (j := j) (t := t)) : Fin (branch j) := | |
| StoppedPathHead.child hij (lowerPath (n := n) rows hij hrows base) | |
| def continuation (base : Base (branch := branch) (i := i) (j := j) (t := t)) : | |
| Path branch j (i + 1) := | |
| StoppedPathHead.tail hij (lowerPath (n := n) rows hij hrows base) | |
| theorem lowerPath_eq_step (base : Base (branch := branch) (i := i) (j := j) (t := t)) : | |
| lowerPath (n := n) rows hij hrows base = | |
| .step (chosen (n := n) rows hij hrows base) (continuation (n := n) rows hij hrows base) := | |
| StoppedPathHead.step_eq hij _ | |
| abbrev projected (base : Base (branch := branch) (i := i) (j := j) (t := t)) := | |
| StoppedProjectedExperiment.projectedSlots clauses rows hupper hij designated o | |
| (key (n := n) rows hrows base) | |
| abbrev projection (base : Base (branch := branch) (i := i) (j := j) (t := t)) := | |
| StoppedProjectedExperiment.projection clauses rows hupper hij designated o | |
| (key (n := n) rows hrows base) | |
| abbrev Exterior (base : Base (branch := branch) (i := i) (j := j) (t := t)) := | |
| StoppedUsefulPairLaw.Descriptor rows repeats (StoppedProjectedExperiment.upperPath hupper o) | |
| (chosen (n := n) rows hij hrows base) (continuation (n := n) rows hij hrows base) | |
| (projected clauses rows hupper hij designated hrows o base) | |
| abbrev Descriptor := (base : Base (branch := branch) (i := i) (j := j) (t := t)) × | |
| Exterior clauses rows repeats hupper hij designated hrows o base | |
| def exteriorLaw (base : Base (branch := branch) (i := i) (j := j) (t := t)) : | |
| FiniteDistribution (Exterior clauses rows repeats hupper hij designated hrows o base) := | |
| StoppedUsefulPairLaw.externalLaw rows repeats (StoppedProjectedExperiment.upperPath hupper o) | |
| (chosen (n := n) rows hij hrows base) (continuation (n := n) rows hij hrows base) | |
| (projected clauses rows hupper hij designated hrows o base) | |
| def externalLaw (hbranch : ∀ k < j + 1, 0 < branch k) | |
| (flag : Fin (branch i) → FiniteDistribution Bool) : | |
| FiniteDistribution (Descriptor clauses rows repeats hupper hij designated hrows o) := | |
| CompletionSoundness.sigmaLaw | |
| (StoppedProjectedBucketIndex.baseLaw (t := t) hij hbranch flag) | |
| (exteriorLaw clauses rows repeats hupper hij designated hrows o) | |
| local instance backgroundFintype : Fintype (HierarchicalMatrixTable.Background (rows := rows) | |
| (StoppedProjectedExperiment.nativeSlots clauses o) (StoppedProjectedExperiment.upper hupper o)) := | |
| Fintype.ofFinite _ | |
| local instance rowSpaceFintype : Fintype (NodeEmbedding.RowSpace | |
| (StoppedProjectedExperiment.nativeSlots clauses o) (StoppedProjectedExperiment.upper hupper o)) := | |
| Fintype.ofFinite _ | |
| def background (descriptor : Descriptor clauses rows repeats hupper hij designated hrows o) : | |
| HierarchicalMatrixTable.Background (rows := rows) | |
| (StoppedProjectedExperiment.nativeSlots clauses o) (StoppedProjectedExperiment.upper hupper o) := | |
| StoppedUsefulPairLaw.background rows repeats (StoppedProjectedExperiment.upperPath hupper o) | |
| (chosen (n := n) rows hij hrows descriptor.1) | |
| (continuation (n := n) rows hij hrows descriptor.1) | |
| (StoppedProjectedExperiment.nativeSlots clauses o) | |
| (projected clauses rows hupper hij designated hrows o descriptor.1) | |
| (projection clauses rows hupper hij designated hrows o descriptor.1) descriptor.2 | |
| def scalarLaw (descriptor : Descriptor clauses rows repeats hupper hij designated hrows o) : | |
| FiniteDistribution (NodeEmbedding.RowSpace | |
| (StoppedProjectedExperiment.nativeSlots clauses o) (StoppedProjectedExperiment.upper hupper o)) := | |
| StoppedUsefulPairLaw.scalarLaw rows repeats (StoppedProjectedExperiment.upperPath hupper o) | |
| (chosen (n := n) rows hij hrows descriptor.1) | |
| (continuation (n := n) rows hij hrows descriptor.1) | |
| (StoppedProjectedExperiment.nativeSlots clauses o) | |
| (projected clauses rows hupper hij designated hrows o descriptor.1) | |
| (projection clauses rows hupper hij designated hrows o descriptor.1) descriptor.2 | |
| variable (hbranch : ∀ k < j + 1, 0 < branch k) | |
| (flag : Fin (branch i) → FiniteDistribution Bool) | |
| (σ : KeyStrategy.Strategy (TreeCanonical.locationCount branch n t)) | |
| abbrev experiment := | |
| StoppedProjectedExperiment.experiment clauses rows repeats hupper hij designated | |
| hbranch hrows flag σ o | |
| include hrows in | |
| theorem rows_positive : 0 < rows (Nodes.height (StoppedProjectedExperiment.upper hupper o)) := by | |
| rw [StoppedProjectedExperiment.upper_height] | |
| exact hrows j | |
| def usefulProbability (κ : ℝ) : ℝ := | |
| HierarchicalAdviceFromBuckets.usefulProbability | |
| (experiment clauses rows repeats hupper hij designated hrows o hbranch flag σ) κ | |
| (externalLaw clauses rows repeats hupper hij designated hrows o hbranch flag) | |
| (background clauses rows repeats hupper hij designated hrows o) | |
| (scalarLaw clauses rows repeats hupper hij designated hrows o) | |
| (rows_positive rows hupper hrows o) | |
| theorem usefulProbability_eq_mean (κ : ℝ) : | |
| usefulProbability clauses rows repeats hupper hij designated hrows o hbranch flag σ κ = | |
| (StoppedProjectedBucketIndex.baseLaw (t := t) hij hbranch flag).expectation (fun base => | |
| HierarchicalAdviceFromBuckets.usefulProbability | |
| (experiment clauses rows repeats hupper hij designated hrows o hbranch flag σ) κ | |
| (exteriorLaw clauses rows repeats hupper hij designated hrows o base) | |
| (fun external => background clauses rows repeats hupper hij designated hrows o | |
| ⟨base, external⟩) | |
| (fun external => scalarLaw clauses rows repeats hupper hij designated hrows o | |
| ⟨base, external⟩) | |
| (rows_positive rows hupper hrows o)) := by | |
| simp only [usefulProbability, HierarchicalAdviceFromBuckets.usefulProbability, | |
| HierarchicalUsefulCollision.law, CompletionSoundness.sigmaLaw_probability, | |
| externalLaw, CandidateCoupling.expectation_sigmaLaw, | |
| HierarchicalUsefulCollision.usefulEvent] | |
| end | |
| end PerfectCompleteness.StoppedProjectedBuckets | |
| end | |
| end OAI | |