openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /CoppersmithWinograd /ComplexCWLocalConstruction.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Polynomial.ComplexPolynomialLocalConstruction | |
| import OAI.LinearAlgebra.MatrixMultiplication.CoppersmithWinograd.ComplexCWOriginalWords | |
| namespace OAI | |
| /-! Coppersmith–Winograd tensors, tensor powers and local restrictions. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.Foundation.CWLocalConstruction | |
| open Tensor OrientationRates CWOriginalWords PolynomialLocalConstruction | |
| abbrev Site (m : ℕ) := Orientation × Fin m | |
| @[simp] theorem site_card (m : ℕ) : Fintype.card (Site m) = 6 * m := by | |
| simp only [Site, Fintype.card_prod, orientation_card, Fintype.card_fin] | |
| def siteEquiv (m : ℕ) : Fin (6 * m) ≃ Site m := | |
| (Fintype.equivFinOfCardEq (site_card m)).symm | |
| def sourceApproximation (m : ℕ) : | |
| PolynomialApproximation (siteTensor (Site := Site m)) | |
| (3 ^ (6 * m)) (2 * (6 * m)) (6 * (6 * m)) := by | |
| have W := (CoppersmithWinograd.approximation.power (6 * m)).pullback | |
| (fun x : Site m → Fin 3 => fun i => x (siteEquiv m i)) | |
| (fun y : Site m → Fin 3 => fun i => y (siteEquiv m i)) | |
| (fun z : Site m → Fin 3 => fun i => z (siteEquiv m i)) | |
| have hsource : Tensor.pullback | |
| (fun x : Site m → Fin 3 => fun i => x (siteEquiv m i)) | |
| (fun y : Site m → Fin 3 => fun i => y (siteEquiv m i)) | |
| (fun z : Site m → Fin 3 => fun i => z (siteEquiv m i)) | |
| (Tensor.power CoppersmithWinograd.tensor (6 * m)) = | |
| siteTensor (Site := Site m) := by | |
| funext x y z | |
| exact (siteTensor_eq_power (siteEquiv m) x y z).symm | |
| rw [hsource] at W | |
| exact W | |
| variable {AX AY AZ Label : Type*} | |
| variable [Fintype AX] [Fintype AY] [Fintype AZ] [Fintype Label] [DecidableEq Label] | |
| variable {m a b c : ℕ} | |
| abbrev CWPrimitive (m a b c : ℕ) := | |
| Primitive (Site m → Fin 3) (Site m → Fin 3) (Site m → Fin 3) | |
| AX AY AZ Label OriginalComponent m a b c | |
| variable (P : CWPrimitive (AX := AX) (AY := AY) (AZ := AZ) (Label := Label) m a b c) | |
| def word (label : Label) : Site m → OriginalComponent := | |
| fun site => P.word label site.1 site.2 | |
| structure SourceCompatible (physical : Site m → Orientation) : Prop where | |
| support_contains : ∀ x y z, | |
| siteTensor x y z ≠ 0 → P.support x y z | |
| grade_decodes : ∀ x y z, siteTensor x y z ≠ 0 → | |
| originalWords (permuteWord physical | |
| (fun site => P.gradeOf x y z site.1 site.2)) = (x, y, z) | |
| designated_coordinates : ∀ label, | |
| (P.designatedX label, P.designatedY label, P.designatedZ label) = | |
| originalWords (permuteWord physical (word P label)) | |
| namespace SourceCompatible | |
| variable {P} {physical : Site m → Orientation} | |
| variable (h : SourceCompatible P physical) | |
| include h | |
| theorem inputSupported : P.InputSupported (siteTensor (Site := Site m)) := by | |
| intro x y z hs | |
| by_contra hn | |
| exact hs (h.support_contains x y z hn) | |
| theorem designatedCoefficients : | |
| P.DesignatedCoefficients (siteTensor (Site := Site m)) := by | |
| intro label | |
| have hc := congrArg (fun xyz => siteTensor xyz.1 xyz.2.1 xyz.2.2) | |
| (h.designated_coordinates label) | |
| exact hc.trans (siteTensor_oriented_originalWords physical (word P label)) | |
| theorem grade_eq_word_iff (x y z : Site m → Fin 3) | |
| (nonzero : siteTensor x y z ≠ 0) (label : Label) : | |
| P.gradeOf x y z = P.word label ↔ | |
| (x, y, z) = (P.designatedX label, P.designatedY label, P.designatedZ label) := by | |
| constructor | |
| · intro hgrade | |
| calc | |
| (x, y, z) = originalWords (permuteWord physical | |
| (fun site => P.gradeOf x y z site.1 site.2)) := | |
| (h.grade_decodes x y z nonzero).symm | |
| _ = originalWords (permuteWord physical (word P label)) := by rw [hgrade]; rfl | |
| _ = (P.designatedX label, P.designatedY label, P.designatedZ label) := | |
| (h.designated_coordinates label).symm | |
| · intro hcoordinates | |
| calc | |
| P.gradeOf x y z = P.gradeOf (P.designatedX label) (P.designatedY label) | |
| (P.designatedZ label) := | |
| congrArg (fun xyz => P.gradeOf xyz.1 xyz.2.1 xyz.2.2) hcoordinates | |
| _ = P.word label := P.designated_word label | |
| def terminalApproximation : | |
| PolynomialApproximation | |
| (Tensor.directSum (fun _ : Label => | |
| matrixCoefficients (K := ℂ) (Fin a) (Fin b) (Fin c))) | |
| (3 ^ (6 * m) * P.rankBound) | |
| ((2 * (6 * m)) * (P.order + 1) + P.order) | |
| ((6 * (6 * m)) * (P.order + 1) + | |
| P.leftDegree + P.middleDegree + P.rightDegree) := | |
| P.terminalApproximation (sourceApproximation m) | |
| h.inputSupported h.designatedCoefficients | |
| end SourceCompatible | |
| structure Admissible (physical : Site m → Orientation) : Prop | |
| extends SourceCompatible P physical where | |
| pairing_budget : a * b * c ≤ P.rankBound | |
| namespace Admissible | |
| variable {P} {physical : Site m → Orientation} | |
| theorem rankBound_pos (h : Admissible P physical) : 0 < P.rankBound := | |
| lt_of_lt_of_le (Nat.mul_pos (Nat.mul_pos P.leftDim_pos P.middleDim_pos) | |
| P.rightDim_pos) h.pairing_budget | |
| end Admissible | |
| end MatrixMultiplication.Foundation.CWLocalConstruction | |
| end | |
| end OAI | |