openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Algebra /HierarchicalMatrixTable.lean
| import OAI.Computability.PerfectCompleteness.Algebra.CanonicalMatrixTable | |
| import OAI.Computability.PerfectCompleteness.Construction.NodeEmbedding | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.HierarchicalMatrixTable | |
| noncomputable section | |
| open scoped Classical | |
| open TreeSourceSpaces HierarchicalArrays | |
| variable {branch rows : Nat → Nat} {n t : Nat} | |
| (slots : RecursiveSpaces.Slots branch n → Fin t → MixedSupport.Slot) | |
| (node : Nodes branch n) | |
| abbrev Background := (j : {j : Nodes branch n // j ≠ node}) → | |
| Fin (rows (Nodes.height j.val)) → NodeEmbedding.NodeH slots j.val | |
| abbrev SideOutput : Type := | |
| BlockQuotient.OtherBlocks (fun j : Nodes branch n => Block rows j) node × PUnit.{1} | |
| abbrev Matrix := Module.Dual F2 (NodeEmbedding.RowSpace slots node) →ₗ[F2] Block rows node | |
| def backgroundOf (arrays : Arrays slots rows) : Background (rows := rows) slots node := | |
| fun j => arrays j.val | |
| def nativeRows (X : Matrix (rows := rows) slots node) : | |
| Fin (rows (Nodes.height node)) → NodeEmbedding.NodeH slots node := | |
| fun i => (NodeEmbedding.equiv slots node).symm | |
| (EvaluationMatrix.rows (NodeEmbedding.RowSpace slots node) X i) | |
| def assemble (background : Background (rows := rows) slots node) | |
| (X : Matrix (rows := rows) slots node) : Arrays slots rows := | |
| fun j => if h : j = node then h.symm ▸ nativeRows slots node X else background ⟨j, h⟩ | |
| @[simp] theorem assemble_selected (background : Background (rows := rows) slots node) | |
| (X : Matrix (rows := rows) slots node) : | |
| assemble slots node background X node = nativeRows slots node X := by | |
| simp [assemble] | |
| @[simp] theorem assemble_other (background : Background (rows := rows) slots node) | |
| (X : Matrix (rows := rows) slots node) (j : {j : Nodes branch n // j ≠ node}) : | |
| assemble slots node background X j.val = background j := by | |
| simp only [assemble, dite_eq_right j.property] | |
| theorem embedded_assembled_rows (background : Background (rows := rows) slots node) | |
| (X : Matrix (rows := rows) slots node) : | |
| NodeEmbedding.embeddedRows (assemble slots node background X) node = | |
| EvaluationMatrix.rows (NodeEmbedding.RowSpace slots node) X := by | |
| funext i | |
| change NodeEmbedding.embed slots node ((assemble slots node background X) node i) = _ | |
| rw [assemble_selected] | |
| exact (NodeEmbedding.equiv slots node).apply_symm_apply _ | |
| theorem matrix_assemble (background : Background (rows := rows) slots node) | |
| (X : Matrix (rows := rows) slots node) : | |
| NodeEmbedding.matrix (assemble slots node background X) node = X := by | |
| unfold NodeEmbedding.matrix | |
| rw [embedded_assembled_rows, EvaluationMatrix.ofRows_rows] | |
| theorem nativeRows_matrix (arrays : Arrays slots rows) : | |
| nativeRows slots node (NodeEmbedding.matrix arrays node) = arrays node := by | |
| funext i | |
| change (NodeEmbedding.equiv slots node).symm | |
| (EvaluationMatrix.rows (NodeEmbedding.RowSpace slots node) | |
| (EvaluationMatrix.ofRows (NodeEmbedding.RowSpace slots node) | |
| (NodeEmbedding.embeddedRows arrays node)) i) = arrays node i | |
| rw [EvaluationMatrix.rows_ofRows] | |
| exact (NodeEmbedding.equiv slots node).symm_apply_apply _ | |
| theorem assemble_original (arrays : Arrays slots rows) : | |
| assemble slots node (backgroundOf slots node arrays) (NodeEmbedding.matrix arrays node) = arrays := by | |
| funext j | |
| by_cases hj : j = node | |
| · subst j | |
| rw [assemble_selected, nativeRows_matrix] | |
| · simp only [assemble, dite_eq_right hj, backgroundOf] | |
| def other (background : Background (rows := rows) slots node) | |
| (x : NodeEmbedding.NumberedDomain slots) : SideOutput (rows := rows) node := | |
| (fun j i => (background j i).val (NodeEmbedding.numberedRestriction slots j.val x), PUnit.unit) | |
| def displayed (background : Background (rows := rows) slots node) | |
| (X : Matrix (rows := rows) slots node) : | |
| NodeEmbedding.NumberedDomain slots → (Output branch n rows : Type) := | |
| TreeCanonical.numberedFunction slots (fullJoint (assemble slots node background X)) | |
| theorem displayed_original (arrays : Arrays slots rows) : | |
| displayed slots node (backgroundOf slots node arrays) (NodeEmbedding.matrix arrays node) = | |
| TreeCanonical.numberedFunction slots (fullJoint arrays) := by | |
| unfold displayed | |
| rw [assemble_original] | |
| theorem query_eq_split_displayed (background : Background (rows := rows) slots node) | |
| (X : Matrix (rows := rows) slots node) : | |
| CanonicalMatrixTable.query (TreeCanonical.numberedSlots slots) | |
| (NodeEmbedding.RowSpace slots node) (other slots node background) X = | |
| BlockQuotient.splitOutput (fun j : Nodes branch n => Block rows j) PUnit.{1} node ∘ | |
| displayed slots node background X := by | |
| funext x | |
| apply Prod.ext | |
| · change EvaluationMatrix.evaluate (NodeEmbedding.RowSpace slots node) X x = | |
| (displayed slots node background X x).1 node | |
| have he := NodeEmbedding.matrix_evaluate (assemble slots node background X) node x | |
| rw [matrix_assemble] at he | |
| exact he | |
| · apply Prod.ext | |
| · funext j i | |
| change (background j i).val (NodeEmbedding.numberedRestriction slots j.val x) = | |
| ((assemble slots node background X) j.val i).val | |
| (NodeEmbedding.numberedRestriction slots j.val x) | |
| rw [assemble_other] | |
| · rfl | |
| variable (σ : KeyStrategy.Strategy (TreeCanonical.locationCount branch n t)) | |
| theorem response_eq_split_response (background : Background (rows := rows) slots node) | |
| (X : Matrix (rows := rows) slots node) : | |
| CanonicalMatrixTable.response (TreeCanonical.numberedSlots slots) | |
| (NodeEmbedding.RowSpace slots node) (other slots node background) σ X = | |
| BlockQuotient.splitOutput (fun j : Nodes branch n => Block rows j) PUnit.{1} node | |
| (KeyStrategy.response σ .left (TreeCanonical.numberedSlots slots) | |
| (displayed slots node background X)) := by | |
| unfold CanonicalMatrixTable.response | |
| rw [query_eq_split_displayed] | |
| exact KeyStrategy.response_postcomp_injective σ .left (TreeCanonical.numberedSlots slots) | |
| (displayed slots node background X) | |
| (BlockQuotient.splitOutput (fun j : Nodes branch n => Block rows j) PUnit.{1} node) | |
| (BlockQuotient.splitOutput (fun j : Nodes branch n => Block rows j) PUnit.{1} node).injective | |
| theorem rightQuery_eq (background : Background (rows := rows) slots node) | |
| (X : Matrix (rows := rows) slots node) (a : Block rows node) : | |
| CanonicalMatrixTable.rightQuery (TreeCanonical.numberedSlots slots) | |
| (NodeEmbedding.RowSpace slots node) (other slots node background) a X = | |
| BlockQuotient.projectBlock node a ∘ displayed slots node background X := by | |
| unfold CanonicalMatrixTable.rightQuery | |
| rw [query_eq_split_displayed] | |
| rfl | |
| theorem accepts_iff_canonical (background : Background (rows := rows) slots node) | |
| (X : Matrix (rows := rows) slots node) (a : Block rows node) : | |
| CanonicalMatrixTable.Accepts (TreeCanonical.numberedSlots slots) | |
| (NodeEmbedding.RowSpace slots node) (other slots node background) σ a X ↔ | |
| CanonicalEdges.coarsen (TreeCanonical.numberedSlots slots) | |
| (displayed slots node background X) (BlockQuotient.projectBlock node a) | |
| (KeyStrategy.label σ .left (TreeCanonical.numberedSlots slots) | |
| (displayed slots node background X)) = | |
| KeyStrategy.label σ .right (TreeCanonical.numberedSlots slots) | |
| (BlockQuotient.projectBlock node a ∘ displayed slots node background X) := by | |
| have hresponse := congrArg (DirectionQuotient.projectWithSide a) | |
| (response_eq_split_response slots node σ background X) | |
| have hright := congrArg | |
| (KeyStrategy.response σ .right (TreeCanonical.numberedSlots slots)) | |
| (rightQuery_eq slots node background X a) | |
| rw [CanonicalMatrixTable.accepts_iff, CanonicalEdges.coarsen_eq_iff] | |
| constructor | |
| · intro h | |
| exact hresponse.symm.trans (h.trans hright) | |
| · intro h | |
| exact hresponse.trans (h.trans hright.symm) | |
| end | |
| end PerfectCompleteness.HierarchicalMatrixTable | |
| end | |
| end OAI | |