openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Algebra /PrefixMatrixAcceptance.lean
| import OAI.Computability.PerfectCompleteness.Algebra.PreliminaryMatrixAcceptance | |
| import OAI.Computability.PerfectCompleteness.Foundations.PrefixTests | |
| import OAI.Computability.PerfectCompleteness.Machines.StoppedOwnInputGeometry | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.PrefixMatrixAcceptance | |
| noncomputable section | |
| open scoped Classical | |
| open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays | |
| variable {branch rows : Nat → Nat} {v m n t : Nat} | |
| (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (σ : KeyStrategy.Strategy (TreeCanonical.locationCount branch n t)) | |
| (questions : PreliminarySampler.Questions branch n t m) | |
| local notation "slots" => sourceSlots clauses (PreliminarySampler.endpoints questions) | |
| theorem observe_iff_matrix | |
| (arrays : Arrays slots rows) (level : Fin n) | |
| (node : PrefixTests.LevelNode branch level) | |
| (a : PrefixTests.LevelDirection rows level) : | |
| PrefixTests.observeAtLevelNode clauses σ questions arrays level node a = true ↔ | |
| CanonicalMatrixTable.Accepts (TreeCanonical.numberedSlots slots) | |
| (NodeEmbedding.RowSpace slots node.val) | |
| (HierarchicalMatrixTable.other slots node.val | |
| (HierarchicalMatrixTable.backgroundOf slots node.val arrays)) | |
| σ (PrefixTests.nodeDirection rows level node a).val | |
| (NodeEmbedding.matrix arrays node.val) := by | |
| have h := (HierarchicalMatrixTable.accepts_iff_canonical slots node.val σ | |
| (HierarchicalMatrixTable.backgroundOf slots node.val arrays) | |
| (NodeEmbedding.matrix arrays node.val) | |
| (PrefixTests.nodeDirection rows level node a).val).symm | |
| rw [HierarchicalMatrixTable.displayed_original] at h | |
| unfold PrefixTests.observeAtLevelNode | |
| dsimp only | |
| constructor | |
| · intro hevent | |
| exact h.mp (@of_decide_eq_true _ _ hevent) | |
| · intro haccept | |
| exact @decide_eq_true _ _ (h.mpr haccept) | |
| theorem observe_iff_lower | |
| (arrays : Arrays slots rows) (level : Fin n) | |
| (node : PrefixTests.LevelNode branch level) | |
| (a : PrefixTests.LevelDirection rows level) (upper : Nodes branch n) : | |
| PrefixTests.observeAtLevelNode clauses σ questions arrays level node a = true ↔ | |
| HierarchicalPrediction.LowerAccepts slots upper | |
| (HierarchicalMatrixTable.backgroundOf slots upper arrays) σ node.val | |
| (PrefixTests.nodeDirection rows level node a).val | |
| (NodeEmbedding.matrix arrays upper) := by | |
| unfold HierarchicalPrediction.LowerAccepts | |
| rw [HierarchicalMatrixTable.displayed_original] | |
| unfold PrefixTests.observeAtLevelNode | |
| dsimp only | |
| constructor | |
| · intro hevent | |
| exact @of_decide_eq_true _ _ hevent | |
| · intro haccept | |
| exact @decide_eq_true _ _ haccept | |
| theorem observe_eq_lower | |
| (arrays : Arrays slots rows) (level : Fin n) | |
| (node : PrefixTests.LevelNode branch level) | |
| (a : PrefixTests.LevelDirection rows level) (upper : Nodes branch n) : | |
| PrefixTests.observeAtLevelNode clauses σ questions arrays level node a = | |
| decide (HierarchicalPrediction.LowerAccepts slots upper | |
| (HierarchicalMatrixTable.backgroundOf slots upper arrays) σ node.val | |
| (PrefixTests.nodeDirection rows level node a).val | |
| (NodeEmbedding.matrix arrays upper)) := by | |
| apply Bool.eq_iff_iff.mpr | |
| simpa only [decide_eq_true_eq] using | |
| observe_iff_lower clauses σ questions arrays level node a upper | |
| theorem observe_endpoint {k : Nat} (p : Path branch n (k + 1)) | |
| (suffix : Slots branch (k + 1)) (arrays : Arrays slots rows) | |
| (a : PrefixTests.LevelDirection rows | |
| (⟨k, Nat.lt_of_lt_of_le (Nat.lt_succ_self k) p.height_le⟩ : Fin n)) : | |
| PreliminaryStrategy.observe clauses σ (questions, p.slotEmbedding suffix) arrays | |
| (PrefixTests.choiceAt rows (p.slotEmbedding suffix) | |
| ⟨k, Nat.lt_of_lt_of_le (Nat.lt_succ_self k) p.height_le⟩ a) = | |
| PrefixTests.observeAtLevelNode clauses σ questions arrays | |
| ⟨k, Nat.lt_of_lt_of_le (Nat.lt_succ_self k) p.height_le⟩ | |
| ⟨WholeArrayInteriorExterior.upperNode p, | |
| WholeArrayInteriorExterior.upperNode_height p⟩ a := by | |
| rw [PrefixTests.observe_choiceAt_eq] | |
| apply congrArg (fun node => PrefixTests.observeAtLevelNode clauses σ questions arrays | |
| ⟨k, Nat.lt_of_lt_of_le (Nat.lt_succ_self k) p.height_le⟩ node a) | |
| apply Subtype.ext | |
| exact StoppedOwnInputGeometry.nodeAtLevel_endpoint p suffix | |
| end | |
| end PerfectCompleteness.PrefixMatrixAcceptance | |
| end | |
| end OAI | |