openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldConstruction /InitialZeroFinish.lean
| import OAI.LinearAlgebra.MatrixMultiplication.CoppersmithWinograd.CWZeroGeometry | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.Tensors | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.InitialLeafRates | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.Terminal | |
| namespace OAI | |
| /-! Tensor extraction over arbitrary fields and its asymptotic rate. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.AllFieldInitialZeroFinish | |
| open MatrixMultiplication.Foundation AllFieldParameters AllFieldHistory AllFieldFiniteFamily | |
| open AllFieldInitialLeafRates CWWindowedLeaves CWStrands CWZeroGeometry | |
| open scoped BigOperators Classical | |
| variable {K : ℕ} | |
| abbrev Index (K : ℕ) := PopulationHistory K | |
| local instance : DecidableEq (Index K) := Classical.decEq _ | |
| theorem shape_total (h : InitialZero K) : shapeTotal (initialShape h.val) = 16 := | |
| (root_shape_spec _ (initialShape_mem h.val)).2 | |
| theorem has_zero (h : InitialZero K) : ∃ i, initialShape h.val i = 0 := | |
| terminal_has_zero (.initial h.val) h.property | |
| theorem canonical_axes_ne (h : InitialZero K) : | |
| zeroAxis (initialShape h.val) ≠ maxAxis (initialShape h.val) := | |
| zero_ne_selected _ _ _ (by rw [shape_total]; norm_num) | |
| (zeroAxis_spec _ (has_zero h)) (maxAxis_spec _) | |
| def zero (h : Index K) : Fin 3 := h.2 (zeroAxis (initialShape h.1.val)) | |
| def selected (h : Index K) : Fin 3 := h.2 (maxAxis (initialShape h.1.val)) | |
| theorem axes_ne (h : Index K) : zero h ≠ selected h := | |
| fun he => canonical_axes_ne h.1 (h.2.injective he) | |
| theorem weights_sum (h : InitialZero K) : | |
| shapeMax (initialShape h.val) + (16 - shapeMax (initialShape h.val)) = 2 * 8 := by | |
| simpa only [shape_total h] using max_add_complement (initialShape h.val) | |
| theorem physical_shape_eq (h : Index K) : | |
| physicalShape h.2 (initialShape h.1.val) = | |
| CWOrientedZero.placedShape (zero h) (selected h) | |
| (shapeMax (initialShape h.1.val)) (16 - shapeMax (initialShape h.1.val)) := by | |
| simpa only [zero, selected, shape_total h.1] using | |
| physicalShape_eq_placedShape (initialShape h.1.val) h.2 | |
| (zeroAxis (initialShape h.1.val)) (maxAxis (initialShape h.1.val)) | |
| (zeroAxis_spec _ (has_zero h.1)) (maxAxis_spec _) (canonical_axes_ne h.1) | |
| variable (allocation : Allocation) (dilation : ℕ) | |
| def embed (h : Index K) (side : Fin 3) (w : Words allocation dilation h) : | |
| HistoryWord allocation dilation (.initial h.1.val, h.2) := | |
| fun i => CWOrientedZero.sideWord (zero h) (selected h) side (w i) | |
| theorem coefficient (F : Type*) [Field F] (ε : ℝ) (h : Index K) | |
| (x y z : Words allocation dilation h) : | |
| historyTensor F allocation dilation ε (.initial h.1.val, h.2) | |
| (embed allocation dilation h 0 x) (embed allocation dilation h 1 y) | |
| (embed allocation dilation h 2 z) = | |
| if CWOrientedZero.matched (zero h) x y z then 1 else 0 := by | |
| rw [historyTensor_initial, physical_shape_eq] | |
| exact CWOrientedZero.power_coefficient F 8 (shapeMax (initialShape h.1.val)) | |
| (16 - shapeMax (initialShape h.1.val)) | |
| (population allocation dilation (.initial h.1.val, h.2)) | |
| (weights_sum h.1) (zero h) (selected h) (axes_ne h) | |
| (fun w : Words allocation dilation h => w) Function.injective_id x y z | |
| abbrev RowIndex (h : Index K) := CWOrientedZero.Row (zero h) (Words allocation dilation h) | |
| abbrev MiddleIndex (h : Index K) := CWOrientedZero.Middle (zero h) (Words allocation dilation h) | |
| abbrev ColumnIndex (h : Index K) := CWOrientedZero.Column (zero h) (Words allocation dilation h) | |
| def factorMap (F : Type*) [Field F] (ε : ℝ) (h : Index K) : | |
| LocalMap (historyTensor F allocation dilation ε (.initial h.1.val, h.2)) | |
| (Tensor.matrixCoefficients (RowIndex allocation dilation h) | |
| (MiddleIndex allocation dilation h) (ColumnIndex allocation dilation h)) := by | |
| letI : Nonempty (Words allocation dilation h) := | |
| Fintype.card_pos_iff.mp (card_words_pos allocation dilation h) | |
| exact CWOrientedZero.ofMatching F _ (zero h) (embed allocation dilation h) | |
| (coefficient allocation dilation F ε h) | |
| theorem factor_volume (h : Index K) : | |
| Fintype.card (RowIndex allocation dilation h) * | |
| Fintype.card (MiddleIndex allocation dilation h) * | |
| Fintype.card (ColumnIndex allocation dilation h) = | |
| Fintype.card (Words allocation dilation h) := | |
| CWOrientedZero.volume (zero h) (Words allocation dilation h) | |
| abbrev RawWords := ∀ h : Index K, | |
| HistoryWord allocation dilation (.initial h.1.val, h.2) | |
| abbrev Rows := ∀ h : Index K, RowIndex allocation dilation h | |
| abbrev Middles := ∀ h : Index K, MiddleIndex allocation dilation h | |
| abbrev Columns := ∀ h : Index K, ColumnIndex allocation dilation h | |
| def source (F : Type*) [Field F] (ε : ℝ) : | |
| Tensor F (RawWords (K := K) allocation dilation) | |
| (RawWords (K := K) allocation dilation) (RawWords (K := K) allocation dilation) := | |
| CommonDimensions.familyProduct (fun h : Index K => | |
| historyTensor F allocation dilation ε (.initial h.1.val, h.2)) | |
| def finishMap (F : Type*) [Field F] (ε : ℝ) : | |
| LocalMap (source (K := K) allocation dilation F ε) | |
| (Tensor.matrixCoefficients (Rows (K := K) allocation dilation) | |
| (Middles (K := K) allocation dilation) (Columns (K := K) allocation dilation)) := | |
| AllFieldTerminal.terminalProductMap | |
| (H := Index K) | |
| (X := fun h => HistoryWord allocation dilation (.initial h.1.val, h.2)) | |
| (Y := fun h => HistoryWord allocation dilation (.initial h.1.val, h.2)) | |
| (Z := fun h => HistoryWord allocation dilation (.initial h.1.val, h.2)) | |
| (fun h : Index K => historyTensor F allocation dilation ε (.initial h.1.val, h.2)) | |
| (RowIndex allocation dilation) (MiddleIndex allocation dilation) | |
| (ColumnIndex allocation dilation) (factorMap allocation dilation F ε) | |
| theorem volume_eq : | |
| Fintype.card (Rows (K := K) allocation dilation) * | |
| Fintype.card (Middles (K := K) allocation dilation) * | |
| Fintype.card (Columns (K := K) allocation dilation) = | |
| AllFieldInitialLeafRates.volume (K := K) allocation dilation := by | |
| rw [AllFieldInitialLeafRates.volume_eq_prod] | |
| simp only [Rows, Middles, Columns, Fintype.card_pi, | |
| ← Finset.prod_mul_distrib] | |
| apply Finset.prod_congr rfl | |
| intro h _ | |
| simpa only [Words, Fintype.card_pi] using factor_volume allocation dilation h | |
| theorem volume_pos : | |
| 0 < Fintype.card (Rows (K := K) allocation dilation) * | |
| Fintype.card (Middles (K := K) allocation dilation) * | |
| Fintype.card (Columns (K := K) allocation dilation) := by | |
| rw [volume_eq] | |
| exact AllFieldInitialLeafRates.volume_pos allocation dilation | |
| theorem rows_pos : 0 < Fintype.card (Rows (K := K) allocation dilation) := | |
| Nat.pos_of_mul_pos_right (Nat.pos_of_mul_pos_right (volume_pos allocation dilation)) | |
| theorem middles_pos : 0 < Fintype.card (Middles (K := K) allocation dilation) := | |
| Nat.pos_of_mul_pos_left (Nat.pos_of_mul_pos_right (volume_pos allocation dilation)) | |
| theorem columns_pos : 0 < Fintype.card (Columns (K := K) allocation dilation) := | |
| Nat.pos_of_mul_pos_left (volume_pos allocation dilation) | |
| end MatrixMultiplication.AllFieldInitialZeroFinish | |
| end | |
| end OAI | |