openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldConstruction /InitialSource.lean
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.FiniteFamily | |
| namespace OAI | |
| /-! Tensor extraction over arbitrary fields and its asymptotic rate. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.AllFieldInitialSource | |
| open MatrixMultiplication.Foundation AllFieldFiniteFamily CommonDimensions | |
| open scoped BigOperators Classical | |
| abbrev Sites (K N : ℕ) := Fin K × (Fin N × Fin 8) | |
| abbrev Words (K N : ℕ) := Fin K → Fin N → Fin 8 → Fin 7 | |
| def sitesEquiv (K N : ℕ) : Sites K N ≃ Fin (8 * K * N) := | |
| Fintype.equivOfCardEq (by simp only [Sites, Fintype.card_prod, Fintype.card_fin]; ring) | |
| def flatten (K N : ℕ) (w : Words K N) : AllFieldSource.Word K N := | |
| fun i => let p := (sitesEquiv K N).symm i; w p.1 p.2.1 p.2.2 | |
| def groupedSource (F : Type*) [Field F] (K N : ℕ) : | |
| Tensor F (Words K N) (Words K N) (Words K N) := | |
| fun x y z => ∏ p : Sites K N, | |
| FieldCW.tensor F 5 (x p.1 p.2.1 p.2.2) (y p.1 p.2.1 p.2.2) (z p.1 p.2.1 p.2.2) | |
| theorem groupedSource_eq_lots (F : Type*) [Field F] (K N : ℕ) : | |
| groupedSource F K N = | |
| familyProduct (fun _ : Fin K => Tensor.power (Tensor.power (FieldCW.tensor F 5) 8) N) := by | |
| funext x y z | |
| simp only [groupedSource, Sites, familyProduct, Tensor.power, Fintype.prod_prod_type] | |
| def groupingMap (F : Type*) [Field F] (K N : ℕ) : | |
| LocalMap (cwSource F K N) (groupedSource F K N) where | |
| x := fun x s => if s = flatten K N x then 1 else 0 | |
| y := fun y s => if s = flatten K N y then 1 else 0 | |
| z := fun z s => if s = flatten K N z then 1 else 0 | |
| coefficient := by | |
| rw [← Tensor.pullback_eq_restrict] | |
| funext x y z | |
| simp only [Tensor.pullback, cwSource, Tensor.power, groupedSource, flatten] | |
| apply Fintype.prod_equiv (sitesEquiv K N).symm | |
| intro i | |
| rfl | |
| def initialExecution (F : Type*) [Field F] (K N : ℕ) : | |
| Execution (cwSource F K N) (groupedSource F K N) := | |
| (startCW F K N).restrict _ (groupingMap F K N) | |
| @[simp] theorem initialExecution_copies (F : Type*) [Field F] (K N : ℕ) : | |
| Fintype.card (initialExecution F K N).Copies = 1 := rfl | |
| end MatrixMultiplication.AllFieldInitialSource | |
| end | |
| end OAI | |