openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Foundations /HierarchicalPrediction.lean
| import OAI.Computability.PerfectCompleteness.Decoding.RepresentativeBucketCollisionLemmas | |
| import OAI.Computability.PerfectCompleteness.Foundations.GoodAdviceEventsLemmas | |
| import OAI.Computability.PerfectCompleteness.Foundations.HierarchicalFrozenTablesLemmas | |
| namespace OAI | |
| namespace PerfectCompleteness.HierarchicalPrediction | |
| noncomputable section | |
| open scoped Classical | |
| open TreeSourceSpaces HierarchicalArrays | |
| open UniqueGamesTheorem.Foundations.Games | |
| open UniqueGamesTheorem.Appendix.RankLevelFilter (linearMapFintype) | |
| attribute [local instance] linearMapFintype | |
| section Selection | |
| variable {E K Y : Type*} [AddCommGroup E] [Module F2 E] | |
| [AddCommGroup K] [Module F2 K] [FiniteDimensional F2 E] | |
| [FiniteDimensional F2 K] [Fintype E] [Fintype K] [Fintype Y] | |
| def selectedValue (f : (E →ₗ[F2] K) → Option Y) (r : Nat) (ρ : ℝ) | |
| (A : ManyGoodRows.RowMap K r) (X : E →ₗ[F2] K) : Option Y := | |
| if h : ManyGoodRows.GoodRow f r ρ A (A.comp X) then | |
| some (ManyGoodRows.selectWitness f r ρ A (A.comp X) h).value | |
| else none | |
| omit [FiniteDimensional F2 E] [FiniteDimensional F2 K] [Fintype Y] in | |
| theorem selectedValue_of_I (f : (E →ₗ[F2] K) → Option Y) (r : Nat) (ρ : ℝ) | |
| (A : ManyGoodRows.RowMap K r) (X : E →ₗ[F2] K) | |
| (hI : GoodAdviceEvents.I f r ρ (A, X) = true) : | |
| ∃ y, selectedValue f r ρ A X = some y ∧ f X = some y := by | |
| unfold GoodAdviceEvents.I GoodAdviceEvents.selectedAgreement at hI | |
| by_cases h : ManyGoodRows.GoodRow f r ρ A (A.comp X) | |
| · simp only [dite_eq_left h, decide_eq_true_eq] at hI | |
| exact ⟨(ManyGoodRows.selectWitness f r ρ A (A.comp X) h).value, | |
| by simp only [selectedValue, dite_eq_left h], hI.2⟩ | |
| · simp only [dite_eq_right h, Bool.false_eq_true] at hI | |
| end Selection | |
| variable {branch rows : Nat → Nat} {n t : Nat} | |
| (slots : RecursiveSpaces.Slots branch n → Fin t → MixedSupport.Slot) | |
| (upper : Nodes branch n) (lowerLevel : Nat) | |
| (background : HierarchicalMatrixTable.Background (rows := rows) slots upper) | |
| (σ : KeyStrategy.Strategy (TreeCanonical.locationCount branch n t)) | |
| abbrev quotientMatrix (X : HierarchicalMatrixTable.Matrix (rows := rows) slots upper) := | |
| MatrixRowQuotient.projectMatrix | |
| (HierarchicalFrozenTables.knownRows slots upper lowerLevel background) X | |
| def affinePrediction (X : HierarchicalMatrixTable.Matrix (rows := rows) slots upper) | |
| (y : Block rows upper × HierarchicalMatrixTable.SideOutput (rows := rows) upper) : | |
| Block rows upper := | |
| X (RepresentativeMatrixTable.correctionFunctional (TreeCanonical.numberedSlots slots) | |
| (NodeEmbedding.RowSpace slots upper) (HierarchicalMatrixTable.other slots upper background) | |
| (HierarchicalFrozenTables.knownRows slots upper lowerLevel background) | |
| (HierarchicalFrozenTables.sectionMap slots upper lowerLevel background) | |
| (HierarchicalFrozenTables.sectionMap_spec slots upper lowerLevel background) y.2) + y.1 | |
| variable (lower : Nodes branch n) (hne : upper ≠ lower) (a : Block rows lower) | |
| def rightUpper (X : HierarchicalMatrixTable.Matrix (rows := rows) slots upper) : | |
| Block rows upper := | |
| (KeyStrategy.response σ .right (TreeCanonical.numberedSlots slots) | |
| (BlockQuotient.projectBlock lower a ∘ | |
| HierarchicalMatrixTable.displayed slots upper background X)).2.1 ⟨upper, hne⟩ | |
| def LowerAccepts (X : HierarchicalMatrixTable.Matrix (rows := rows) slots upper) : Prop := | |
| CanonicalEdges.coarsen (TreeCanonical.numberedSlots slots) | |
| (HierarchicalMatrixTable.displayed slots upper background X) | |
| (BlockQuotient.projectBlock lower a) | |
| (KeyStrategy.label σ .left (TreeCanonical.numberedSlots slots) | |
| (HierarchicalMatrixTable.displayed slots upper background X)) = | |
| KeyStrategy.label σ .right (TreeCanonical.numberedSlots slots) | |
| (BlockQuotient.projectBlock lower a ∘ | |
| HierarchicalMatrixTable.displayed slots upper background X) | |
| theorem rightUpper_eq_raw (X : HierarchicalMatrixTable.Matrix (rows := rows) slots upper) | |
| (hacc : LowerAccepts slots upper background σ lower a X) : | |
| rightUpper slots upper background σ lower hne a X = | |
| (CanonicalMatrixTable.response (TreeCanonical.numberedSlots slots) | |
| (NodeEmbedding.RowSpace slots upper) (HierarchicalMatrixTable.other slots upper background) | |
| σ X).1 := by | |
| unfold LowerAccepts at hacc | |
| rw [CanonicalEdges.coarsen_eq_iff] at hacc | |
| have hcoordinate := congrArg (fun z => z.2.1 ⟨upper, hne⟩) hacc | |
| have hraw := congrArg Prod.fst | |
| (HierarchicalMatrixTable.response_eq_split_response slots upper σ background X) | |
| exact hcoordinate.symm.trans hraw.symm | |
| variable {Ω : Type*} [Fintype Ω] | |
| (original : FiniteDistribution Ω) (arrays : Ω → Arrays slots rows) | |
| (lowerEvent : Ω → Bool) (κ : ℝ) | |
| theorem table_some_iff (X : HierarchicalMatrixTable.Matrix (rows := rows) slots upper) | |
| (y : Block rows upper × HierarchicalMatrixTable.SideOutput (rows := rows) upper) : | |
| HierarchicalUsefulness.table slots upper lowerLevel original arrays lowerEvent κ σ background | |
| (quotientMatrix slots upper lowerLevel background X) = some y ↔ | |
| HierarchicalUsefulness.mark slots upper lowerLevel original arrays lowerEvent κ | |
| ⟨background, quotientMatrix slots upper lowerLevel background X⟩ = true ∧ | |
| RepresentativeMatrixTable.response (TreeCanonical.numberedSlots slots) | |
| (NodeEmbedding.RowSpace slots upper) (HierarchicalMatrixTable.other slots upper background) | |
| σ (HierarchicalFrozenTables.knownRows slots upper lowerLevel background) | |
| (HierarchicalFrozenTables.sectionMap slots upper lowerLevel background) X = y := by | |
| unfold HierarchicalUsefulness.table HierarchicalFrozenTables.table | |
| RepresentativeMatrixTable.partialTable RepresentativeMatrixTable.response | |
| RepresentativeMatrixTable.representative | |
| split <;> simp_all | |
| theorem I_implies_mark (r : Nat) (ρ : ℝ) | |
| (A : ManyGoodRows.RowMap (Block rows upper) r) | |
| (X : HierarchicalMatrixTable.Matrix (rows := rows) slots upper) | |
| (hI : GoodAdviceEvents.I | |
| (HierarchicalUsefulness.table slots upper lowerLevel original arrays lowerEvent κ σ background) | |
| r ρ (A, quotientMatrix slots upper lowerLevel background X) = true) : | |
| HierarchicalUsefulness.mark slots upper lowerLevel original arrays lowerEvent κ | |
| ⟨background, quotientMatrix slots upper lowerLevel background X⟩ = true := by | |
| obtain ⟨y, _, hy⟩ := selectedValue_of_I _ r ρ A _ hI | |
| exact ((table_some_iff slots upper lowerLevel background σ original arrays lowerEvent κ X y).mp hy).1 | |
| def prediction (r : Nat) (ρ : ℝ) (A : ManyGoodRows.RowMap (Block rows upper) r) | |
| (X : HierarchicalMatrixTable.Matrix (rows := rows) slots upper) : Bool := | |
| let f := HierarchicalUsefulness.table slots upper lowerLevel original arrays lowerEvent κ σ background | |
| let Q := quotientMatrix slots upper lowerLevel background X | |
| GoodAdviceEvents.J f r ρ (A, Q) && | |
| match selectedValue f r ρ A Q with | |
| | none => false | |
| | some y => decide (rightUpper slots upper background σ lower hne a X = | |
| affinePrediction slots upper lowerLevel background X y) | |
| theorem lower_accepts_predicts (r : Nat) (ρ : ℝ) | |
| (A : ManyGoodRows.RowMap (Block rows upper) r) | |
| (X : HierarchicalMatrixTable.Matrix (rows := rows) slots upper) | |
| (hacc : LowerAccepts slots upper background σ lower a X) | |
| (hI : GoodAdviceEvents.I | |
| (HierarchicalUsefulness.table slots upper lowerLevel original arrays lowerEvent κ σ background) | |
| r ρ (A, quotientMatrix slots upper lowerLevel background X) = true) : | |
| prediction slots upper lowerLevel background σ lower hne a original arrays lowerEvent κ r ρ A X = true := by | |
| obtain ⟨y, hselected, hy⟩ := selectedValue_of_I _ r ρ A _ hI | |
| have hmatch := ((table_some_iff slots upper lowerLevel background σ original arrays | |
| lowerEvent κ X y).mp hy).2 | |
| have hraw := RepresentativeDecoder.raw_selected_output_at_match | |
| (TreeCanonical.numberedSlots slots) (NodeEmbedding.RowSpace slots upper) | |
| (HierarchicalMatrixTable.other slots upper background) | |
| (HierarchicalFrozenTables.knownRows slots upper lowerLevel background) | |
| (HierarchicalFrozenTables.sectionMap slots upper lowerLevel background) | |
| (HierarchicalFrozenTables.sectionMap_spec slots upper lowerLevel background) σ | |
| (HierarchicalFrozenTables.determined slots upper lowerLevel background) X y hmatch | |
| have hright : rightUpper slots upper background σ lower hne a X = | |
| affinePrediction slots upper lowerLevel background X y := | |
| (rightUpper_eq_raw slots upper background σ lower hne a X hacc).trans hraw | |
| have hJ := GoodAdviceEvents.I_implies_J _ r ρ _ hI | |
| simp only [prediction, hJ, hselected, hright, decide_true, Bool.and_true] | |
| end | |
| end PerfectCompleteness.HierarchicalPrediction | |
| end OAI | |