openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Construction /NodeEmbedding.lean
| import Mathlib.LinearAlgebra.Dimension.Finrank | |
| import OAI.Computability.PerfectCompleteness.Algebra.EvaluationMatrix | |
| import OAI.Computability.PerfectCompleteness.Algebra.UniformLinearImage | |
| import OAI.Computability.PerfectCompleteness.Foundations.HierarchicalArrays | |
| namespace OAI | |
| namespace PerfectCompleteness.NodeEmbedding | |
| open TreeSourceSpaces HierarchicalArrays | |
| open UniqueGamesTheorem.Foundations.Games | |
| open scoped Classical | |
| noncomputable section | |
| variable {branch : Nat → Nat} {n t : Nat} | |
| abbrev NumberedDomain (slots : RecursiveSpaces.Slots branch n → Fin t → MixedSupport.Slot) := | |
| MixedSupport.Assignment (TreeCanonical.numberedSlots slots) | |
| abbrev NodeH (slots : RecursiveSpaces.Slots branch n → Fin t → MixedSupport.Slot) | |
| (node : Nodes branch n) := H (nodeSlots slots node) | |
| variable (slots : RecursiveSpaces.Slots branch n → Fin t → MixedSupport.Slot) | |
| (node : Nodes branch n) | |
| def numberedRestriction : NumberedDomain slots → Domain (nodeSlots slots node) := | |
| restrictNode slots node ∘ (TreeCanonical.assignmentEquiv slots).symm | |
| theorem numberedRestriction_surjective : Function.Surjective (numberedRestriction slots node) := | |
| (restrictNode_surjective slots node).comp (TreeCanonical.assignmentEquiv slots).symm.surjective | |
| theorem pullback_injective : | |
| Function.Injective (PointwiseSpaces.pullback F2 (numberedRestriction slots node)) := | |
| PointwiseSpaces.pullback_injective _ (numberedRestriction_surjective slots node) | |
| def RowSpace : Submodule F2 (NumberedDomain slots → F2) := | |
| (NodeH slots node).map (PointwiseSpaces.pullback F2 (numberedRestriction slots node)) | |
| def equiv : NodeH slots node ≃ₗ[F2] RowSpace slots node := | |
| Submodule.equivMapOfInjective | |
| (PointwiseSpaces.pullback F2 (numberedRestriction slots node)) | |
| (pullback_injective slots node) (NodeH slots node) | |
| def embed : NodeH slots node →ₗ[F2] RowSpace slots node := | |
| (equiv slots node).toLinearMap | |
| @[simp] theorem equiv_apply (f : NodeH slots node) (x : NumberedDomain slots) : | |
| (equiv slots node f).val x = f.val (numberedRestriction slots node x) := rfl | |
| @[simp] theorem embed_apply (f : NodeH slots node) (x : NumberedDomain slots) : | |
| (embed slots node f).val x = f.val (numberedRestriction slots node x) := rfl | |
| theorem embed_injective : Function.Injective (embed slots node) := | |
| (equiv slots node).injective | |
| theorem embed_surjective : Function.Surjective (embed slots node) := | |
| (equiv slots node).surjective | |
| theorem finrank_eq : | |
| Module.finrank F2 (RowSpace slots node) = Module.finrank F2 (NodeH slots node) := | |
| (equiv slots node).finrank_eq.symm | |
| theorem card_eq : Nat.card (RowSpace slots node) = Nat.card (NodeH slots node) := | |
| (Nat.card_congr (equiv slots node).toEquiv).symm | |
| theorem uniform_embed [Fintype (NodeH slots node)] [Fintype (RowSpace slots node)] : | |
| (FiniteDistribution.uniform (NodeH slots node)).pushforward (embed slots node) = | |
| FiniteDistribution.uniform (RowSpace slots node) := | |
| UniformLinearImage.uniform_pushforward_linearMap (embed slots node) | |
| (embed_surjective slots node) | |
| def rowsEquiv (ℓ : Nat) : | |
| (Fin ℓ → NodeH slots node) ≃ₗ[F2] (Fin ℓ → RowSpace slots node) := | |
| LinearEquiv.piCongrRight (fun _ : Fin ℓ => equiv slots node) | |
| @[simp] theorem rowsEquiv_apply (ℓ : Nat) (f : Fin ℓ → NodeH slots node) (i : Fin ℓ) : | |
| rowsEquiv slots node ℓ f i = embed slots node (f i) := rfl | |
| theorem uniform_rowsEquiv (ℓ : Nat) | |
| [Fintype (NodeH slots node)] [Fintype (RowSpace slots node)] : | |
| (FiniteDistribution.uniform (Fin ℓ → NodeH slots node)).pushforward | |
| (rowsEquiv slots node ℓ) = | |
| FiniteDistribution.uniform (Fin ℓ → RowSpace slots node) := | |
| UniformLinearImage.uniform_pushforward_linearMap | |
| (rowsEquiv slots node ℓ).toLinearMap (rowsEquiv slots node ℓ).surjective | |
| theorem squareSpace_eq : | |
| PointwiseSpaces.squareSpace (RowSpace slots node) = | |
| (PointwiseSpaces.squareSpace (NodeH slots node)).map | |
| (PointwiseSpaces.pullback F2 (numberedRestriction slots node)) := | |
| (PointwiseSpaces.squareSpace_map_pullback | |
| (numberedRestriction slots node) (NodeH slots node)).symm | |
| theorem embed_mul (f g : NodeH slots node) : | |
| (embed slots node f).val * (embed slots node g).val = | |
| PointwiseSpaces.pullback F2 (numberedRestriction slots node) (f.val * g.val) := rfl | |
| def RootRowSpace : Submodule F2 (NumberedDomain slots → F2) := | |
| (H slots).map (PointwiseSpaces.pullback F2 (TreeCanonical.assignmentEquiv slots).symm) | |
| theorem rowSpace_le_root (hbranch : ∀ k < n, 0 < branch k) : | |
| RowSpace slots node ≤ RootRowSpace slots := by | |
| intro f hf | |
| obtain ⟨g, hg, rfl⟩ := Submodule.mem_map.mp hf | |
| have hroot : PointwiseSpaces.pullback F2 (restrictNode slots node) g ∈ H slots := | |
| DescendantSpaces.space_descendant_le (Nodes.path node) (LeafDomain slots) hbranch | |
| (Submodule.mem_map_of_mem hg) | |
| exact Submodule.mem_map.mpr | |
| ⟨PointwiseSpaces.pullback F2 (restrictNode slots node) g, hroot, rfl⟩ | |
| theorem squareSpace_le_root (hbranch : ∀ k < n, 0 < branch k) | |
| (hproper : Nodes.height node < n) : | |
| PointwiseSpaces.squareSpace (RowSpace slots node) ≤ RootRowSpace slots := by | |
| rw [squareSpace_eq] | |
| intro f hf | |
| obtain ⟨g, hg, rfl⟩ := Submodule.mem_map.mp hf | |
| have hroot : PointwiseSpaces.pullback F2 (restrictNode slots node) g ∈ H slots := | |
| DescendantSpaces.descendant_square_le_space (Nodes.path node) (LeafDomain slots) | |
| hbranch hproper (Submodule.mem_map_of_mem hg) | |
| exact Submodule.mem_map.mpr | |
| ⟨PointwiseSpaces.pullback F2 (restrictNode slots node) g, hroot, rfl⟩ | |
| variable {slots} {rows : Nat → Nat} | |
| def embeddedRows (arrays : Arrays slots rows) (node : Nodes branch n) : | |
| Fin (rows (Nodes.height node)) → RowSpace slots node := | |
| fun i => embed slots node (arrays node i) | |
| @[simp] theorem embeddedRows_apply (arrays : Arrays slots rows) (node : Nodes branch n) | |
| (i : Fin (rows (Nodes.height node))) (x : NumberedDomain slots) : | |
| (embeddedRows arrays node i).val x = | |
| (TreeCanonical.numberedFunction slots (fullJoint arrays) x).1 node i := rfl | |
| def matrix (arrays : Arrays slots rows) (node : Nodes branch n) : | |
| Module.Dual F2 (RowSpace slots node) →ₗ[F2] Block rows node := | |
| EvaluationMatrix.ofRows (RowSpace slots node) (embeddedRows arrays node) | |
| @[simp] theorem matrix_evaluate (arrays : Arrays slots rows) (node : Nodes branch n) | |
| (x : NumberedDomain slots) : | |
| EvaluationMatrix.evaluate (RowSpace slots node) (matrix arrays node) x = | |
| (TreeCanonical.numberedFunction slots (fullJoint arrays) x).1 node := rfl | |
| theorem evaluation_comp_embed (x : NumberedDomain slots) : | |
| (EvaluationMatrix.evaluation (RowSpace slots node) x).comp (embed slots node) = | |
| EvaluationMatrix.evaluation (NodeH slots node) (numberedRestriction slots node x) := by | |
| ext f | |
| rfl | |
| end | |
| end PerfectCompleteness.NodeEmbedding | |
| end OAI | |