openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Algebra /PreliminaryMatrixAcceptance.lean
| import OAI.Computability.PerfectCompleteness.Foundations.HierarchicalPrediction | |
| import OAI.Computability.PerfectCompleteness.Reduction.HierarchicalFixedAdviceFamilyLemmas | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.PreliminaryMatrixAcceptance | |
| noncomputable section | |
| open scoped Classical | |
| open PreliminarySampler TreeSourceSpaces HierarchicalArrays | |
| variable {v m n t : Nat} {branch rows : Nat → Nat} | |
| (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (σ : KeyStrategy.Strategy (TreeCanonical.locationCount branch n t)) | |
| (b : Base branch n t m) | |
| theorem observe_iff_matrix_accepts | |
| (arrays : Arrays (sourceSlots clauses (endpoints b.1)) rows) | |
| (choice : Choice rows b.2) : | |
| PreliminaryStrategy.observe clauses σ b arrays choice = true ↔ | |
| CanonicalMatrixTable.Accepts | |
| (TreeCanonical.numberedSlots (sourceSlots clauses (endpoints b.1))) | |
| (NodeEmbedding.RowSpace (sourceSlots clauses (endpoints b.1)) | |
| (GeometricPath.nodeAtLevel b.2 choice.1)) | |
| (HierarchicalMatrixTable.other (sourceSlots clauses (endpoints b.1)) | |
| (GeometricPath.nodeAtLevel b.2 choice.1) | |
| (HierarchicalMatrixTable.backgroundOf (sourceSlots clauses (endpoints b.1)) | |
| (GeometricPath.nodeAtLevel b.2 choice.1) arrays)) | |
| σ choice.2.val (NodeEmbedding.matrix arrays (GeometricPath.nodeAtLevel b.2 choice.1)) := by | |
| have h := (HierarchicalMatrixTable.accepts_iff_canonical | |
| (sourceSlots clauses (endpoints b.1)) (GeometricPath.nodeAtLevel b.2 choice.1) σ | |
| (HierarchicalMatrixTable.backgroundOf (sourceSlots clauses (endpoints b.1)) | |
| (GeometricPath.nodeAtLevel b.2 choice.1) arrays) | |
| (NodeEmbedding.matrix arrays (GeometricPath.nodeAtLevel b.2 choice.1)) choice.2.val).symm | |
| rw [HierarchicalMatrixTable.displayed_original] at h | |
| unfold PreliminaryStrategy.observe | |
| dsimp only | |
| constructor | |
| · intro hobserve | |
| exact h.mp (@of_decide_eq_true _ _ hobserve) | |
| · intro haccepts | |
| exact @decide_eq_true _ _ (h.mpr haccepts) | |
| theorem observe_eq_decide_matrix_accepts | |
| (arrays : Arrays (sourceSlots clauses (endpoints b.1)) rows) | |
| (choice : Choice rows b.2) : | |
| PreliminaryStrategy.observe clauses σ b arrays choice = | |
| decide (CanonicalMatrixTable.Accepts | |
| (TreeCanonical.numberedSlots (sourceSlots clauses (endpoints b.1))) | |
| (NodeEmbedding.RowSpace (sourceSlots clauses (endpoints b.1)) | |
| (GeometricPath.nodeAtLevel b.2 choice.1)) | |
| (HierarchicalMatrixTable.other (sourceSlots clauses (endpoints b.1)) | |
| (GeometricPath.nodeAtLevel b.2 choice.1) | |
| (HierarchicalMatrixTable.backgroundOf (sourceSlots clauses (endpoints b.1)) | |
| (GeometricPath.nodeAtLevel b.2 choice.1) arrays)) | |
| σ choice.2.val (NodeEmbedding.matrix arrays (GeometricPath.nodeAtLevel b.2 choice.1))) := by | |
| apply Bool.eq_iff_iff.mpr | |
| simpa only [decide_eq_true_eq] using observe_iff_matrix_accepts clauses σ b arrays choice | |
| theorem observe_assembled_iff_lower_accepts (upper : Nodes branch n) | |
| (background : HierarchicalMatrixTable.Background (rows := rows) | |
| (sourceSlots clauses (endpoints b.1)) upper) | |
| (X : HierarchicalMatrixTable.Matrix (rows := rows) (sourceSlots clauses (endpoints b.1)) upper) | |
| (choice : Choice rows b.2) : | |
| PreliminaryStrategy.observe clauses σ b | |
| (HierarchicalMatrixTable.assemble (sourceSlots clauses (endpoints b.1)) upper background X) | |
| choice = true ↔ | |
| HierarchicalPrediction.LowerAccepts (sourceSlots clauses (endpoints b.1)) upper | |
| background σ (GeometricPath.nodeAtLevel b.2 choice.1) choice.2.val X := by | |
| unfold PreliminaryStrategy.observe | |
| dsimp only | |
| constructor | |
| · intro hobserve | |
| have hcanonical := @of_decide_eq_true _ _ hobserve | |
| exact hcanonical | |
| · intro haccepts | |
| exact @decide_eq_true _ _ haccepts | |
| theorem observe_iff_lower_accepts | |
| (arrays : Arrays (sourceSlots clauses (endpoints b.1)) rows) | |
| (choice : Choice rows b.2) (upper : Nodes branch n) : | |
| PreliminaryStrategy.observe clauses σ b arrays choice = true ↔ | |
| HierarchicalPrediction.LowerAccepts (sourceSlots clauses (endpoints b.1)) upper | |
| (HierarchicalMatrixTable.backgroundOf (sourceSlots clauses (endpoints b.1)) upper arrays) | |
| σ (GeometricPath.nodeAtLevel b.2 choice.1) choice.2.val (NodeEmbedding.matrix arrays upper) := by | |
| have h := observe_assembled_iff_lower_accepts clauses σ b upper | |
| (HierarchicalMatrixTable.backgroundOf (sourceSlots clauses (endpoints b.1)) upper arrays) | |
| (NodeEmbedding.matrix arrays upper) choice | |
| rwa [HierarchicalMatrixTable.assemble_original] at h | |
| end | |
| end PerfectCompleteness.PreliminaryMatrixAcceptance | |
| end | |
| end OAI | |