openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Decoding /HierarchicalLeftDecoder.lean
| import OAI.Computability.PerfectCompleteness.Algebra.FunctionSpaceLowerRank | |
| import OAI.Computability.PerfectCompleteness.Decoding.LeftDecoderKnownRows | |
| import OAI.Computability.PerfectCompleteness.Foundations.HierarchicalFrozenTablesLemmas | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.HierarchicalLeftDecoder | |
| noncomputable section | |
| open scoped Classical | |
| open TreeSourceSpaces HierarchicalArrays PointwiseSpaces | |
| open UniqueGamesTheorem.Foundations.Games | |
| attribute [local instance] UniqueGamesTheorem.Appendix.RankLevelFilter.linearMapFintype | |
| variable {branch rows : Nat → Nat} {n t : Nat} | |
| (slots : RecursiveSpaces.Slots branch n → Fin t → MixedSupport.Slot) | |
| (upper : Nodes branch n) (lowerLevel : Nat) | |
| abbrev LowerNode (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) := | |
| (HierarchicalFrozenTables.lowerIndex upper lowerLevel d).val | |
| 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 _ | |
| local instance upperDualFintype : Fintype (Module.Dual F2 (NodeEmbedding.RowSpace slots upper)) := | |
| LeftDecoder.dualFintype (V := NodeEmbedding.RowSpace slots upper) | |
| variable (hbranch : ∀ k < n, 0 < branch k) | |
| def squareEmbedding (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) : | |
| squareSpace (NodeEmbedding.NodeH slots (LowerNode upper lowerLevel d)) →ₗ[F2] | |
| NodeEmbedding.RowSpace slots upper where | |
| toFun f := ⟨PointwiseSpaces.pullback F2 | |
| (NodeEmbedding.numberedRestriction slots (LowerNode upper lowerLevel d)) f.val, by | |
| apply RelativeDescendantProducts.squareSpace_le_rowSpace slots hbranch | |
| (RelativeDescendantProducts.descendantIndex_below upper d.val) | |
| rw [NodeEmbedding.squareSpace_eq] | |
| exact Submodule.mem_map_of_mem f.property⟩ | |
| map_add' _ _ := Subtype.ext rfl | |
| map_smul' _ _ := Subtype.ext rfl | |
| @[simp] theorem squareEmbedding_apply | |
| (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) | |
| (f : squareSpace (NodeEmbedding.NodeH slots (LowerNode upper lowerLevel d))) | |
| (x : NodeEmbedding.NumberedDomain slots) : | |
| (squareEmbedding slots upper lowerLevel hbranch d f).val x = | |
| f.val (NodeEmbedding.numberedRestriction slots (LowerNode upper lowerLevel d) x) := rfl | |
| def output (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) | |
| (q : Module.Dual F2 (NodeEmbedding.RowSpace slots upper)) : | |
| Module.Dual F2 (squareSpace (NodeEmbedding.NodeH slots (LowerNode upper lowerLevel d))) := | |
| q.comp (squareEmbedding slots upper lowerLevel hbranch d) | |
| def lowerOne (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) : | |
| squareSpace (NodeEmbedding.NodeH slots (LowerNode upper lowerLevel d)) := | |
| ⟨1, TreeSourceSpaces.one_mem_squareSpace | |
| (nodeSlots slots (LowerNode upper lowerLevel d)) | |
| (fun k hk => hbranch k (Nat.lt_of_lt_of_le hk | |
| (Nodes.height_le (LowerNode upper lowerLevel d))))⟩ | |
| variable (background : HierarchicalMatrixTable.Background (rows := rows) slots upper) | |
| def testedMatrix (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) : | |
| Module.Dual F2 (NodeEmbedding.NodeH slots (LowerNode upper lowerLevel d)) →ₗ[F2] | |
| Block rows (LowerNode upper lowerLevel d) := | |
| EvaluationMatrix.ofRows (NodeEmbedding.NodeH slots (LowerNode upper lowerLevel d)) | |
| (background (HierarchicalFrozenTables.lowerIndex upper lowerLevel d)) | |
| def oneKnown : HierarchicalFrozenTables.knownRows slots upper lowerLevel background := | |
| ⟨⟨1, RelativeDescendantProducts.one_mem_rowSpace slots hbranch upper⟩, | |
| HierarchicalKnownRows.one_mem_knownSpace slots upper background | |
| (HierarchicalFrozenTables.lowerIndex upper lowerLevel)⟩ | |
| def productKnown (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) | |
| (i j : Fin (rows (Nodes.height (LowerNode upper lowerLevel d)))) : | |
| HierarchicalFrozenTables.knownRows slots upper lowerLevel background := | |
| ⟨squareEmbedding slots upper lowerLevel hbranch d | |
| (OddListExtraction.productElement | |
| (NodeEmbedding.NodeH slots (LowerNode upper lowerLevel d)) | |
| (background (HierarchicalFrozenTables.lowerIndex upper lowerLevel d) i) | |
| (background (HierarchicalFrozenTables.lowerIndex upper lowerLevel d) j)), | |
| HierarchicalKnownRows.product_mem_knownSpace slots upper background | |
| (HierarchicalFrozenTables.lowerIndex upper lowerLevel) d i j⟩ | |
| theorem output_normalized_of_known_realizer | |
| (q : Module.Dual F2 (NodeEmbedding.RowSpace slots upper)) | |
| (x : NodeEmbedding.NumberedDomain slots) | |
| (hknown : ∀ b : HierarchicalFrozenTables.knownRows slots upper lowerLevel background, | |
| q (b : NodeEmbedding.RowSpace slots upper) = b.val.val x) | |
| (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) : | |
| output slots upper lowerLevel hbranch d q (lowerOne slots upper lowerLevel hbranch d) = 1 := by | |
| exact hknown (oneKnown slots upper lowerLevel hbranch background) | |
| theorem tested_rank_le_one_of_known_realizer | |
| (q : Module.Dual F2 (NodeEmbedding.RowSpace slots upper)) | |
| (x : NodeEmbedding.NumberedDomain slots) | |
| (hknown : ∀ b : HierarchicalFrozenTables.knownRows slots upper lowerLevel background, | |
| q (b : NodeEmbedding.RowSpace slots upper) = b.val.val x) | |
| (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) : | |
| (LowerAffineSliceRank.testedGram | |
| (FunctionSpaceLowerRank.matrixForm | |
| (NodeEmbedding.NodeH slots (LowerNode upper lowerLevel d)) | |
| (OddListExtraction.multiplicationForm | |
| (NodeEmbedding.NodeH slots (LowerNode upper lowerLevel d)) | |
| (output slots upper lowerLevel hbranch d q))) | |
| (testedMatrix slots upper lowerLevel background d)).rank ≤ 1 := by | |
| let v := HierarchicalKnownRows.lower slots upper background | |
| (HierarchicalFrozenTables.lowerIndex upper lowerLevel) x d | |
| have hgram : LowerAffineSliceRank.testedGram | |
| (FunctionSpaceLowerRank.matrixForm | |
| (NodeEmbedding.NodeH slots (LowerNode upper lowerLevel d)) | |
| (OddListExtraction.multiplicationForm | |
| (NodeEmbedding.NodeH slots (LowerNode upper lowerLevel d)) | |
| (output slots upper lowerLevel hbranch d q))) | |
| (testedMatrix slots upper lowerLevel background d) = fun i j => v i * v j := by | |
| rw [FunctionSpaceLowerRank.testedGram_multiplicationForm] | |
| simp only [testedMatrix, EvaluationMatrix.rows_ofRows] | |
| funext i j | |
| exact hknown (productKnown slots upper lowerLevel hbranch background d i j) | |
| rw [hgram] | |
| exact Matrix.rank_vecMulVec_le v v | |
| def Valid (q : Module.Dual F2 (NodeEmbedding.RowSpace slots upper)) : Prop := | |
| ∀ d : HierarchicalFrozenTables.LowerNodes upper lowerLevel, | |
| output slots upper lowerLevel hbranch d q (lowerOne slots upper lowerLevel hbranch d) = 1 ∧ | |
| (LowerAffineSliceRank.testedGram | |
| (FunctionSpaceLowerRank.matrixForm | |
| (NodeEmbedding.NodeH slots (LowerNode upper lowerLevel d)) | |
| (OddListExtraction.multiplicationForm | |
| (NodeEmbedding.NodeH slots (LowerNode upper lowerLevel d)) | |
| (output slots upper lowerLevel hbranch d q))) | |
| (testedMatrix slots upper lowerLevel background d)).rank ≤ 1 | |
| theorem valid_of_known_realizer | |
| (q : Module.Dual F2 (NodeEmbedding.RowSpace slots upper)) | |
| (hreal : ∃ x : NodeEmbedding.NumberedDomain slots, | |
| ∀ b : HierarchicalFrozenTables.knownRows slots upper lowerLevel background, | |
| q (b : NodeEmbedding.RowSpace slots upper) = b.val.val x) : | |
| Valid slots upper lowerLevel hbranch background q := by | |
| obtain ⟨x, hx⟩ := hreal | |
| intro d | |
| exact ⟨output_normalized_of_known_realizer slots upper lowerLevel hbranch background q x hx d, | |
| tested_rank_le_one_of_known_realizer slots upper lowerLevel hbranch background q x hx d⟩ | |
| variable (σ : KeyStrategy.Strategy (TreeCanonical.locationCount branch n t)) | |
| (useful : HierarchicalFrozenTables.QuotientMatrix slots upper lowerLevel background → Prop) | |
| def adviceLaw (r : Nat) (ρ : ℝ) (A : ManyGoodRows.RowMap (Block rows upper) r) | |
| (U : Module.Dual F2 (NodeEmbedding.RowSpace slots upper ⧸ | |
| HierarchicalFrozenTables.knownRows slots upper lowerLevel background) →ₗ[F2] | |
| (Fin r → F2)) : FiniteDistribution (Module.Dual F2 (NodeEmbedding.RowSpace slots upper)) := | |
| LeftDecoder.adviceLaw (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) | |
| (HierarchicalFrozenTables.sectionMap_spec slots upper lowerLevel background) useful r ρ A U | |
| theorem adviceLaw_valid (r : Nat) (ρ : ℝ) (hρ : 0 < ρ) | |
| (A : ManyGoodRows.RowMap (Block rows upper) r) | |
| (U : Module.Dual F2 (NodeEmbedding.RowSpace slots upper ⧸ | |
| HierarchicalFrozenTables.knownRows slots upper lowerLevel background) →ₗ[F2] | |
| (Fin r → F2)) : | |
| (adviceLaw slots upper lowerLevel background σ useful r ρ A U).probability | |
| (fun q => decide (Valid slots upper lowerLevel hbranch background q)) = 1 := by | |
| unfold adviceLaw | |
| apply LeftDecoderKnownRows.adviceLaw_probability_eq_one_of_known_realizer | |
| (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) | |
| (HierarchicalFrozenTables.sectionMap_spec slots upper lowerLevel background) useful | |
| (HierarchicalFrozenTables.determined slots upper lowerLevel background) r ρ hρ A U | |
| intro q hq | |
| exact decide_eq_true (valid_of_known_realizer slots upper lowerLevel hbranch background q hq) | |
| variable {r : Nat} {ρ : ℝ} {A : ManyGoodRows.RowMap (Block rows upper) r} | |
| {U : Module.Dual F2 (NodeEmbedding.RowSpace slots upper ⧸ | |
| HierarchicalFrozenTables.knownRows slots upper lowerLevel background) →ₗ[F2] | |
| (Fin r → F2)} | |
| theorem candidate_valid | |
| (w : DenseWitnessRealization.RepresentativeWitness | |
| (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) useful | |
| (r := r) (ρ := ρ) (A := A) (U := U)) | |
| (hρ : 0 < ρ) | |
| (q : PulledColumnSpace.Candidates | |
| (HierarchicalFrozenTables.knownRows slots upper lowerLevel background) w.columnSpace | |
| (LeftDecoder.center (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) | |
| (HierarchicalFrozenTables.sectionMap_spec slots upper lowerLevel background) useful w)) : | |
| Valid slots upper lowerLevel hbranch background q.val := by | |
| obtain ⟨x, _, hx⟩ := LeftDecoder.exists_candidate_realizer | |
| (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) | |
| (HierarchicalFrozenTables.sectionMap_spec slots upper lowerLevel background) useful | |
| (HierarchicalFrozenTables.determined slots upper lowerLevel background) w hρ | |
| exact valid_of_known_realizer slots upper lowerLevel hbranch background q.val ⟨x, hx q⟩ | |
| end | |
| end PerfectCompleteness.HierarchicalLeftDecoder | |
| end | |
| end OAI | |