openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldConstruction /InitialState.lean
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.InitialHistories | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.Tensors | |
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.CanonicalInitial | |
| namespace OAI | |
| /-! Tensor extraction over arbitrary fields and its asymptotic rate. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.AllFieldInitialState | |
| open MatrixMultiplication.Foundation AllFieldHistory AllFieldParameters AllFieldFiniteFamily | |
| open scoped BigOperators Classical | |
| variable {K : ℕ} (allocation : Allocation) (dilation : ℕ) | |
| abbrev CanonicalKey := JointCanonicalization.ClassKey (H := Fin K) | |
| abbrev PlacementPositions (p : InitialPlacement K) := | |
| Fin (population allocation dilation (.initial p.1, p.2)) | |
| def classPosition (c : CanonicalKey (K := K)) | |
| (i : Fin (initialJointCounts allocation dilation c.1 c.2)) : | |
| Σ p : InitialPlacement K, PlacementPositions allocation dilation p := | |
| let q := initialClassPositionEquiv allocation dilation c.1 c.2 i | |
| ⟨((c.1, q.1.val.1), q.1.val.2), q.2⟩ | |
| theorem classPosition_shape (c : CanonicalKey (K := K)) | |
| (i : Fin (initialJointCounts allocation dilation c.1 c.2)) : | |
| JointCanonicalCW.shapeNat c.2 = | |
| physicalShape (classPosition allocation dilation c i).1.2 | |
| (initialShape (classPosition allocation dilation c i).1.1) := by | |
| have h := congrArg decodeShape (initialClassPosition_shape allocation dilation c.1 c.2 i) | |
| simp only [rootShape, decode_encodePhysicalShape] at h | |
| exact h.symm | |
| theorem prod_classPosition {F : Type*} [CommMonoid F] | |
| (f : ∀ p : InitialPlacement K, PlacementPositions allocation dilation p → F) : | |
| (∏ c : CanonicalKey (K := K), | |
| ∏ i : Fin (initialJointCounts allocation dilation c.1 c.2), | |
| f (classPosition allocation dilation c i).1 | |
| (classPosition allocation dilation c i).2) = | |
| ∏ p : InitialPlacement K, ∏ i : PlacementPositions allocation dilation p, f p i := by | |
| conv_lhs => rw [Fintype.prod_prod_type] | |
| conv_rhs => | |
| rw [Fintype.prod_prod_type] | |
| rw [Fintype.prod_prod_type] | |
| apply Finset.prod_congr rfl | |
| intro j _ | |
| calc | |
| (∏ u : JointPopulation.Shape, ∏ i : Fin (initialJointCounts allocation dilation j u), | |
| f (classPosition allocation dilation (j, u) i).1 | |
| (classPosition allocation dilation (j, u) i).2) = | |
| ∏ u : JointPopulation.Shape, ∏ p : InitialShapeFiber j u, | |
| ∏ i : Fin (population allocation dilation (.initial (j, p.val.1), p.val.2)), | |
| f ((j, p.val.1), p.val.2) i := by | |
| apply Finset.prod_congr rfl | |
| intro u _ | |
| rw [← Fintype.prod_sigma'] | |
| apply Fintype.prod_equiv (initialClassPositionEquiv allocation dilation j u) | |
| intro i | |
| rfl | |
| _ = ∏ p : Fin sortedInitial.length × Placement, | |
| ∏ i : Fin (population allocation dilation (.initial (j, p.1), p.2)), | |
| f ((j, p.1), p.2) i := | |
| Fintype.prod_fiberwise (fun p : Fin sortedInitial.length × Placement => | |
| rootShape (j, p.1) p.2) | |
| (fun p => ∏ i : Fin (population allocation dilation (.initial (j, p.1), p.2)), | |
| f ((j, p.1), p.2) i) | |
| _ = _ := by rw [Fintype.prod_prod_type] | |
| abbrev StateWords := ∀ h : State K 0, HistoryWord allocation dilation h.val | |
| def toCanonical (w : StateWords (K := K) allocation dilation) : | |
| JointCanonicalInitial.CanonicalWords (initialJointCounts (K := K) allocation dilation) := | |
| fun c i => | |
| let q := classPosition allocation dilation c i | |
| w ((initialStateEquiv K).symm q.1) q.2 | |
| theorem coefficient (F : Type*) [Field F] (ε : ℝ) | |
| (x y z : StateWords (K := K) allocation dilation) : | |
| JointCanonicalInitial.canonical (F := F) (initialJointCounts allocation dilation) | |
| (toCanonical allocation dilation x) (toCanonical allocation dilation y) | |
| (toCanonical allocation dilation z) = | |
| stateTensor F allocation dilation ε 0 x y z := by | |
| let f : ∀ p : InitialPlacement K, PlacementPositions allocation dilation p → F := | |
| fun p i => CWStrands.shapeTensor (Fin 8) (physicalShape p.2 (initialShape p.1)) | |
| (x ((initialStateEquiv K).symm p) i) | |
| (y ((initialStateEquiv K).symm p) i) | |
| (z ((initialStateEquiv K).symm p) i) | |
| have hcanonical : | |
| JointCanonicalInitial.canonical (F := F) (initialJointCounts allocation dilation) | |
| (toCanonical allocation dilation x) (toCanonical allocation dilation y) | |
| (toCanonical allocation dilation z) = | |
| ∏ p : InitialPlacement K, ∏ i : PlacementPositions allocation dilation p, f p i := by | |
| rw [← prod_classPosition allocation dilation f] | |
| unfold JointCanonicalInitial.canonical InheritedMasks.classProduct | |
| apply Finset.prod_congr rfl | |
| intro c _ | |
| apply Finset.prod_congr rfl | |
| intro i _ | |
| dsimp only [toCanonical, f] | |
| rw [classPosition_shape allocation dilation c i] | |
| have hstate : stateTensor F allocation dilation ε 0 x y z = | |
| ∏ p : InitialPlacement K, ∏ i : PlacementPositions allocation dilation p, f p i := by | |
| change (∏ h : State K 0, historyTensor F allocation dilation ε h.val (x h) (y h) (z h)) = _ | |
| rw [← (initialStateEquiv K).symm.prod_comp | |
| (fun h : State K 0 => historyTensor F allocation dilation ε h.val (x h) (y h) (z h))] | |
| apply Finset.prod_congr rfl | |
| intro p _ | |
| change historyTensor F allocation dilation ε (.initial p.1, p.2) | |
| (x ((initialStateEquiv K).symm p)) (y ((initialStateEquiv K).symm p)) | |
| (z ((initialStateEquiv K).symm p)) = _ | |
| rw [historyTensor_initial] | |
| rfl | |
| exact hcanonical.trans hstate.symm | |
| def map (F : Type*) [Field F] (ε : ℝ) : | |
| LocalMap | |
| (JointCanonicalInitial.canonical (F := F) (initialJointCounts (K := K) allocation dilation)) | |
| (stateTensor F (K := K) allocation dilation ε 0) where | |
| x := fun x s => if s = toCanonical allocation dilation x then 1 else 0 | |
| y := fun y s => if s = toCanonical allocation dilation y then 1 else 0 | |
| z := fun z s => if s = toCanonical allocation dilation z then 1 else 0 | |
| coefficient := by | |
| rw [← Tensor.pullback_eq_restrict] | |
| funext x y z | |
| exact coefficient allocation dilation F ε x y z | |
| def execution (F : Type*) [Field F] (ε : ℝ) {I : Type} [Fintype I] {N : ℕ} | |
| (E : Execution (cwSource F K N) | |
| (Tensor.directSum (fun _ : I => JointCanonicalInitial.canonical | |
| (initialJointCounts (K := K) allocation dilation)))) : | |
| Execution (cwSource F K N) | |
| (Tensor.directSum (fun _ : I => stateTensor F (K := K) allocation dilation ε 0)) := | |
| E.restrict _ ((map (K := K) allocation dilation F ε).parallelCopies I) | |
| @[simp] theorem execution_copies (F : Type*) [Field F] (ε : ℝ) | |
| {I : Type} [Fintype I] {N : ℕ} | |
| (E : Execution (cwSource F K N) | |
| (Tensor.directSum (fun _ : I => JointCanonicalInitial.canonical | |
| (initialJointCounts (K := K) allocation dilation)))) : | |
| Fintype.card (execution allocation dilation F ε E).Copies = Fintype.card E.Copies := rfl | |
| end MatrixMultiplication.AllFieldInitialState | |
| end | |
| end OAI | |