openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Decoding /HierarchicalAllDecoderTables.lean
| import Mathlib.Data.Fintype.Sigma | |
| import OAI.Computability.PerfectCompleteness.Decoding.HierarchicalDecoderTables | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.HierarchicalAllDecoderTables | |
| noncomputable section | |
| open scoped Classical | |
| open TreeSourceSpaces HierarchicalArrays PointwiseSpaces | |
| open UniqueGamesTheorem.Foundations.Games | |
| attribute [local instance] UniqueGamesTheorem.Appendix.RankLevelFilter.linearMapFintype | |
| private theorem row_probability_of_raw {Q A : Type*} [Fintype A] | |
| (μ : FiniteDistribution (Option A)) (valid : Q → A → Prop) (q : Q) | |
| (predicate : A → Prop) (hp : valid q = predicate) | |
| (hv : μ.probability (fun answer => decide | |
| (∀ z, answer = some z → predicate z)) = 1) : | |
| μ.probability (fun answer => decide | |
| (DecoderTableAdmissibility.RowValid valid q answer)) = 1 := by | |
| have hevent : (fun answer : Option A => decide | |
| (DecoderTableAdmissibility.RowValid valid q answer)) = | |
| (fun answer : Option A => decide (∀ z, answer = some z → predicate z)) := by | |
| funext answer | |
| apply (@decide_eq_decide | |
| (DecoderTableAdmissibility.RowValid valid q answer) | |
| (∀ z, answer = some z → predicate z) _ _).mpr | |
| unfold DecoderTableAdmissibility.RowValid | |
| rw [hp] | |
| rw [hevent] | |
| exact hv | |
| variable {branch rows : Nat → Nat} {n t : Nat} | |
| (slots : RecursiveSpaces.Slots branch n → Fin t → MixedSupport.Slot) | |
| (upper : Nodes branch n) (lowerLevel : Nat) | |
| abbrev Background := HierarchicalMatrixTable.Background (rows := rows) slots upper | |
| abbrev Input (r : Nat) := | |
| (background : Background (rows := rows) slots upper) × | |
| HierarchicalDecoderTables.Advice slots upper lowerLevel background r | |
| local instance rowSpaceFintype (node : Nodes branch n) : | |
| Fintype (NodeEmbedding.RowSpace slots node) := Fintype.ofFinite _ | |
| local instance nodeSpaceFintype (node : Nodes branch n) : | |
| Fintype (NodeEmbedding.NodeH slots node) := Fintype.ofFinite _ | |
| instance inputFintype (r : Nat) : Fintype (Input (rows := rows) slots upper lowerLevel r) := by | |
| letI : Fintype (Background (rows := rows) slots upper) := Fintype.ofFinite _ | |
| letI : (background : Background (rows := rows) slots upper) → | |
| Fintype (HierarchicalDecoderTables.Advice slots upper lowerLevel background r) := | |
| fun _ => inferInstance | |
| exact inferInstanceAs (Fintype | |
| ((background : Background (rows := rows) slots upper) × | |
| HierarchicalDecoderTables.Advice slots upper lowerLevel background r)) | |
| abbrev UpperAnswer := Module.Dual F2 (NodeEmbedding.RowSpace slots upper) | |
| local instance upperDualFintype : Fintype (UpperAnswer slots upper) := | |
| LeftDecoder.dualFintype (V := NodeEmbedding.RowSpace slots upper) | |
| local instance answerFintype (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) : | |
| Fintype (HierarchicalDecoderTables.Answer slots upper lowerLevel d) := | |
| LeftDecoder.dualFintype | |
| (V := squareSpace (HierarchicalDecoderTables.LowerH slots upper lowerLevel d)) | |
| variable (hbranch : ∀ k < n, 0 < branch k) | |
| (σ : KeyStrategy.Strategy (TreeCanonical.locationCount branch n t)) | |
| (useful : (background : Background (rows := rows) slots upper) → | |
| HierarchicalFrozenTables.QuotientMatrix slots upper lowerLevel background → Prop) | |
| def upperKernel (r : Nat) (ρ : ℝ) (input : Input (rows := rows) slots upper lowerLevel r) : | |
| FiniteDistribution (UpperAnswer slots upper) := | |
| HierarchicalLeftDecoder.adviceLaw slots upper lowerLevel input.1 σ (useful input.1) | |
| r ρ input.2.1 input.2.2 | |
| def upperTableLaw (r : Nat) (ρ : ℝ) : | |
| FiniteDistribution (Input (rows := rows) slots upper lowerLevel r → UpperAnswer slots upper) := | |
| FiniteDistribution.table (upperKernel slots upper lowerLevel σ useful r ρ) | |
| def decodeTable (r : Nat) | |
| (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) (s : Nat) | |
| (table : Input (rows := rows) slots upper lowerLevel r → UpperAnswer slots upper) : | |
| Input (rows := rows) slots upper lowerLevel r → | |
| Option (HierarchicalDecoderTables.Answer slots upper lowerLevel d) := | |
| fun input => HierarchicalDecoderTables.decode slots upper lowerLevel hbranch d s (table input) | |
| def responseLaw (r : Nat) (ρ : ℝ) | |
| (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) (s : Nat) | |
| (input : Input (rows := rows) slots upper lowerLevel r) : | |
| FiniteDistribution (Option (HierarchicalDecoderTables.Answer slots upper lowerLevel d)) := | |
| (upperKernel slots upper lowerLevel σ useful r ρ input).pushforward | |
| (HierarchicalDecoderTables.decode slots upper lowerLevel hbranch d s) | |
| def tableLaw (r : Nat) (ρ : ℝ) | |
| (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) (s : Nat) : | |
| FiniteDistribution (Input (rows := rows) slots upper lowerLevel r → | |
| Option (HierarchicalDecoderTables.Answer slots upper lowerLevel d)) := | |
| FiniteDistribution.table (responseLaw slots upper lowerLevel hbranch σ useful r ρ d s) | |
| theorem upperTableLaw_pushforward (r : Nat) (ρ : ℝ) | |
| (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) (s : Nat) : | |
| (upperTableLaw slots upper lowerLevel σ useful r ρ).pushforward | |
| (decodeTable slots upper lowerLevel hbranch r d s) = | |
| tableLaw slots upper lowerLevel hbranch σ useful r ρ d s := by | |
| exact CleanConditioning.pushforward_law | |
| (I := Input (rows := rows) slots upper lowerLevel r) | |
| (Ω := fun _ => UpperAnswer slots upper) | |
| (Γ := fun _ => Option (HierarchicalDecoderTables.Answer slots upper lowerLevel d)) | |
| (upperKernel slots upper lowerLevel σ useful r ρ) | |
| (fun _ => HierarchicalDecoderTables.decode slots upper lowerLevel hbranch d s) | |
| theorem supported_upper_table_valid (r : Nat) (ρ : ℝ) (hρ : 0 < ρ) | |
| (table : Input (rows := rows) slots upper lowerLevel r → UpperAnswer slots upper) | |
| (hsupport : (upperTableLaw slots upper lowerLevel σ useful r ρ).weight table ≠ 0) | |
| (input : Input (rows := rows) slots upper lowerLevel r) : | |
| HierarchicalLeftDecoder.Valid slots upper lowerLevel hbranch input.1 (table input) := by | |
| have hrow := FiniteDistribution.table_row_weight_ne_zero | |
| (upperKernel slots upper lowerLevel σ useful r ρ) table hsupport input | |
| exact of_decide_eq_true (DecoderTableAdmissibility.event_of_probability_one | |
| (upperKernel slots upper lowerLevel σ useful r ρ input) | |
| (fun q => decide (HierarchicalLeftDecoder.Valid slots upper lowerLevel hbranch input.1 q)) | |
| (HierarchicalLeftDecoder.adviceLaw_valid slots upper lowerLevel hbranch input.1 | |
| σ (useful input.1) r ρ hρ input.2.1 input.2.2) (table input) hrow) | |
| theorem upperTableLaw_valid (r : Nat) (ρ : ℝ) (hρ : 0 < ρ) : | |
| (upperTableLaw slots upper lowerLevel σ useful r ρ).probability | |
| (fun table => decide (∀ input : Input (rows := rows) slots upper lowerLevel r, | |
| HierarchicalLeftDecoder.Valid slots upper lowerLevel hbranch input.1 (table input))) = 1 := by | |
| apply DecoderTableAdmissibility.probability_one_of_support | |
| intro table hsupport | |
| exact decide_eq_true (supported_upper_table_valid slots upper lowerLevel hbranch σ useful | |
| r ρ hρ table hsupport) | |
| def RowValid (r : Nat) (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) (s : Nat) | |
| (input : Input (rows := rows) slots upper lowerLevel r) | |
| (z : HierarchicalDecoderTables.Answer slots upper lowerLevel d) : Prop := | |
| HierarchicalDecoderTables.Valid slots upper lowerLevel input.1 hbranch d s z | |
| theorem responseLaw_valid (r : Nat) (ρ : ℝ) (hρ : 0 < ρ) | |
| (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) (s : Nat) | |
| (input : Input (rows := rows) slots upper lowerLevel r) : | |
| (responseLaw slots upper lowerLevel hbranch σ useful r ρ d s input).probability | |
| (fun answer => decide (DecoderTableAdmissibility.RowValid | |
| (RowValid slots upper lowerLevel hbranch r d s) input answer)) = 1 := by | |
| have hlaw : responseLaw slots upper lowerLevel hbranch σ useful r ρ d s input = | |
| HierarchicalDecoderTables.responseLaw slots upper lowerLevel input.1 hbranch | |
| σ (useful input.1) r ρ d s input.2 := rfl | |
| rw [hlaw] | |
| apply row_probability_of_raw | |
| (Q := Input (rows := rows) slots upper lowerLevel r) | |
| (A := HierarchicalDecoderTables.Answer slots upper lowerLevel d) | |
| (HierarchicalDecoderTables.responseLaw slots upper lowerLevel input.1 hbranch | |
| σ (useful input.1) r ρ d s input.2) | |
| (RowValid slots upper lowerLevel hbranch r d s) input | |
| (HierarchicalDecoderTables.Valid slots upper lowerLevel input.1 hbranch d s) rfl | |
| rw [← HierarchicalDecoderTables.responseLaw_valid slots upper lowerLevel input.1 hbranch σ (useful input.1) r ρ hρ d s input.2] | |
| apply congrArg (HierarchicalDecoderTables.responseLaw slots upper lowerLevel input.1 hbranch | |
| σ (useful input.1) r ρ d s input.2).probability | |
| funext answer | |
| exact (@decide_eq_decide _ _ _ _).mpr Iff.rfl | |
| theorem supported_table_valid (r : Nat) (ρ : ℝ) (hρ : 0 < ρ) | |
| (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) (s : Nat) | |
| (table : Input (rows := rows) slots upper lowerLevel r → | |
| Option (HierarchicalDecoderTables.Answer slots upper lowerLevel d)) | |
| (hsupport : (tableLaw slots upper lowerLevel hbranch σ useful r ρ d s).weight table ≠ 0) : | |
| DecoderTableAdmissibility.Admissible (RowValid slots upper lowerLevel hbranch r d s) table := by | |
| exact DecoderTableAdmissibility.table_admissible_of_probability_one | |
| (responseLaw slots upper lowerLevel hbranch σ useful r ρ d s) | |
| (RowValid slots upper lowerLevel hbranch r d s) | |
| (responseLaw_valid slots upper lowerLevel hbranch σ useful r ρ hρ d s) table hsupport | |
| theorem tableLaw_valid (r : Nat) (ρ : ℝ) (hρ : 0 < ρ) | |
| (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) (s : Nat) : | |
| (tableLaw slots upper lowerLevel hbranch σ useful r ρ d s).probability | |
| (fun table => decide (DecoderTableAdmissibility.Admissible | |
| (RowValid slots upper lowerLevel hbranch r d s) table)) = 1 := | |
| DecoderTableAdmissibility.table_admissible_probability_one | |
| (responseLaw slots upper lowerLevel hbranch σ useful r ρ d s) | |
| (RowValid slots upper lowerLevel hbranch r d s) | |
| (responseLaw_valid slots upper lowerLevel hbranch σ useful r ρ hρ d s) | |
| theorem decoded_supported_table_valid (r : Nat) (ρ : ℝ) (hρ : 0 < ρ) | |
| (table : Input (rows := rows) slots upper lowerLevel r → UpperAnswer slots upper) | |
| (hsupport : (upperTableLaw slots upper lowerLevel σ useful r ρ).weight table ≠ 0) | |
| (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) (s : Nat) : | |
| DecoderTableAdmissibility.Admissible (RowValid slots upper lowerLevel hbranch r d s) | |
| (decodeTable slots upper lowerLevel hbranch r d s table) := by | |
| intro input z hz | |
| exact HierarchicalDecoderTables.decode_valid slots upper lowerLevel input.1 hbranch d s | |
| (table input) (supported_upper_table_valid slots upper lowerLevel hbranch σ useful | |
| r ρ hρ table hsupport input) z hz | |
| end | |
| end PerfectCompleteness.HierarchicalAllDecoderTables | |
| end | |
| end OAI | |