openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Sampling /HierarchicalArrayLaw.lean
| import OAI.Computability.PerfectCompleteness.Algebra.HierarchicalMatrixTable | |
| import OAI.Computability.PerfectCompleteness.Foundations.QuarterBalanceLemmas | |
| import OAI.Computability.PerfectCompleteness.Sampling.BucketUniformLemmas | |
| import OAI.Computability.PerfectCompleteness.Sampling.WholeArraySampler | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.HierarchicalArrayLaw | |
| noncomputable section | |
| open scoped Classical | |
| open TreeSourceSpaces HierarchicalArrays | |
| open UniqueGamesTheorem.Foundations.Games | |
| variable {branch : Nat → Nat} {n t : Nat} | |
| (slots : RecursiveSpaces.Slots branch n → Fin t → MixedSupport.Slot) | |
| (rows : Nat → Nat) | |
| abbrev NativeRows (node : Nodes branch n) := | |
| Fin (rows (Nodes.height node)) → NodeEmbedding.NodeH slots node | |
| def law (μ : (node : Nodes branch n) → FiniteDistribution (NativeRows slots rows node)) : | |
| FiniteDistribution (Arrays slots rows) := | |
| FiniteProduct.law μ | |
| def otherLaw (μ : (node : Nodes branch n) → FiniteDistribution (NativeRows slots rows node)) | |
| (node : Nodes branch n) : | |
| FiniteDistribution (HierarchicalMatrixTable.Background (rows := rows) slots node) := | |
| FiniteProductSplit.otherLaw μ node | |
| def nativeToMatrix (node : Nodes branch n) (v : NativeRows slots rows node) : | |
| HierarchicalMatrixTable.Matrix (rows := rows) slots node := | |
| EvaluationMatrix.ofRows (NodeEmbedding.RowSpace slots node) | |
| (NodeEmbedding.rowsEquiv slots node (rows (Nodes.height node)) v) | |
| @[simp] theorem nativeToMatrix_selected (arrays : Arrays slots rows) (node : Nodes branch n) : | |
| nativeToMatrix slots rows node (arrays node) = NodeEmbedding.matrix arrays node := rfl | |
| private theorem pushforward_id {A : Type*} [Fintype A] (μ : FiniteDistribution A) : | |
| μ.pushforward (fun a => a) = μ := by | |
| apply FiniteDistribution.eq_of_weight_eq | |
| intro a | |
| simp [FiniteDistribution.pushforward] | |
| theorem matrix_background_law | |
| (μ : (node : Nodes branch n) → FiniteDistribution (NativeRows slots rows node)) | |
| (node : Nodes branch n) | |
| [Fintype (HierarchicalMatrixTable.Matrix (rows := rows) slots node)] : | |
| (law slots rows μ).pushforward (fun arrays => | |
| (NodeEmbedding.matrix arrays node, HierarchicalMatrixTable.backgroundOf slots node arrays)) = | |
| ((μ node).pushforward (nativeToMatrix slots rows node)).product | |
| (otherLaw slots rows μ node) := by | |
| have h := congrArg | |
| (fun ν : FiniteDistribution (NativeRows slots rows node × | |
| HierarchicalMatrixTable.Background (rows := rows) slots node) => | |
| ν.pushforward (fun p => (nativeToMatrix slots rows node p.1, p.2))) | |
| (FiniteProductSplit.split_law μ node) | |
| rw [FiniteDistribution.pushforward_comp] at h | |
| have hp := FiniteDistribution.product_pushforward | |
| (μ node) (FiniteProductSplit.otherLaw μ node) (nativeToMatrix slots rows node) | |
| (id : HierarchicalMatrixTable.Background (rows := rows) slots node → | |
| HierarchicalMatrixTable.Background (rows := rows) slots node) | |
| have hi : (FiniteProductSplit.otherLaw μ node).pushforward | |
| (id : HierarchicalMatrixTable.Background (rows := rows) slots node → | |
| HierarchicalMatrixTable.Background (rows := rows) slots node) = | |
| FiniteProductSplit.otherLaw μ node := pushforward_id (FiniteProductSplit.otherLaw μ node) | |
| rw [hi] at hp | |
| exact h.trans (by simpa only [id_eq, otherLaw] using hp) | |
| theorem uniform_nativeToMatrix (node : Nodes branch n) | |
| [Fintype (HierarchicalMatrixTable.Matrix (rows := rows) slots node)] : | |
| (FiniteDistribution.uniform (NativeRows slots rows node)).pushforward | |
| (nativeToMatrix slots rows node) = | |
| FiniteDistribution.uniform (HierarchicalMatrixTable.Matrix (rows := rows) slots node) := by | |
| let : Fintype (NodeEmbedding.RowSpace slots node) := Fintype.ofFinite _ | |
| calc | |
| _ = ((FiniteDistribution.uniform (NativeRows slots rows node)).pushforward | |
| (NodeEmbedding.rowsEquiv slots node (rows (Nodes.height node)))).pushforward | |
| (EvaluationMatrix.ofRows (NodeEmbedding.RowSpace slots node)) := | |
| (FiniteDistribution.pushforward_comp _ _ _).symm | |
| _ = (FiniteDistribution.uniform | |
| (Fin (rows (Nodes.height node)) → NodeEmbedding.RowSpace slots node)).pushforward | |
| (EvaluationMatrix.ofRows (NodeEmbedding.RowSpace slots node)) := by | |
| rw [NodeEmbedding.uniform_rowsEquiv] | |
| _ = _ := UniformLinearImage.uniform_pushforward_linearMap | |
| (BucketUniform.ofRowsLinear (NodeEmbedding.RowSpace slots node) (rows (Nodes.height node))) | |
| (BucketUniform.ofRowsLinear_surjective | |
| (NodeEmbedding.RowSpace slots node) (rows (Nodes.height node))) | |
| theorem uniform_matrix_background_law | |
| (μ : (node : Nodes branch n) → FiniteDistribution (NativeRows slots rows node)) | |
| (node : Nodes branch n) | |
| [Fintype (HierarchicalMatrixTable.Matrix (rows := rows) slots node)] | |
| (hselected : μ node = FiniteDistribution.uniform (NativeRows slots rows node)) : | |
| (law slots rows μ).pushforward (fun arrays => | |
| (NodeEmbedding.matrix arrays node, HierarchicalMatrixTable.backgroundOf slots node arrays)) = | |
| (FiniteDistribution.uniform | |
| (HierarchicalMatrixTable.Matrix (rows := rows) slots node)).product | |
| (otherLaw slots rows μ node) := by | |
| rw [matrix_background_law, hselected, uniform_nativeToMatrix] | |
| theorem matrix_coarse_background_law | |
| (μ : (node : Nodes branch n) → FiniteDistribution (NativeRows slots rows node)) | |
| (node : Nodes branch n) | |
| [Fintype (HierarchicalMatrixTable.Matrix (rows := rows) slots node)] | |
| {B : Type*} [Fintype B] | |
| (record : HierarchicalMatrixTable.Background (rows := rows) slots node → B) : | |
| (law slots rows μ).pushforward (fun arrays => | |
| (NodeEmbedding.matrix arrays node, record (HierarchicalMatrixTable.backgroundOf slots node arrays))) = | |
| ((μ node).pushforward (nativeToMatrix slots rows node)).product | |
| ((otherLaw slots rows μ node).pushforward record) := by | |
| have h := congrArg | |
| (fun ν : FiniteDistribution | |
| (HierarchicalMatrixTable.Matrix (rows := rows) slots node × | |
| HierarchicalMatrixTable.Background (rows := rows) slots node) => | |
| ν.pushforward (fun p => (p.1, record p.2))) | |
| (matrix_background_law slots rows μ node) | |
| rw [FiniteDistribution.pushforward_comp] at h | |
| have hp := FiniteDistribution.product_pushforward | |
| ((μ node).pushforward (nativeToMatrix slots rows node)) (otherLaw slots rows μ node) | |
| (id : HierarchicalMatrixTable.Matrix (rows := rows) slots node → | |
| HierarchicalMatrixTable.Matrix (rows := rows) slots node) record | |
| have hi : ((μ node).pushforward (nativeToMatrix slots rows node)).pushforward | |
| (id : HierarchicalMatrixTable.Matrix (rows := rows) slots node → | |
| HierarchicalMatrixTable.Matrix (rows := rows) slots node) = | |
| (μ node).pushforward (nativeToMatrix slots rows node) := | |
| pushforward_id ((μ node).pushforward (nativeToMatrix slots rows node)) | |
| rw [hi] at hp | |
| exact h.trans (by simpa only [id_eq] using hp) | |
| theorem uniform_matrix_coarse_background_law | |
| (μ : (node : Nodes branch n) → FiniteDistribution (NativeRows slots rows node)) | |
| (node : Nodes branch n) | |
| [Fintype (HierarchicalMatrixTable.Matrix (rows := rows) slots node)] | |
| (hselected : μ node = FiniteDistribution.uniform (NativeRows slots rows node)) | |
| {B : Type*} [Fintype B] | |
| (record : HierarchicalMatrixTable.Background (rows := rows) slots node → B) : | |
| (law slots rows μ).pushforward (fun arrays => | |
| (NodeEmbedding.matrix arrays node, record (HierarchicalMatrixTable.backgroundOf slots node arrays))) = | |
| (FiniteDistribution.uniform | |
| (HierarchicalMatrixTable.Matrix (rows := rows) slots node)).product | |
| ((otherLaw slots rows μ node).pushforward record) := by | |
| rw [matrix_coarse_background_law, hselected, uniform_nativeToMatrix] | |
| end | |
| end PerfectCompleteness.HierarchicalArrayLaw | |
| end | |
| end OAI | |