| import OAI.LinearAlgebra.MatrixMultiplication.Completion.RecursiveCompletion |
|
|
| namespace OAI |
|
|
| /-! Readable tensor completion and its finite arithmetic realization. -/ |
|
|
| noncomputable section |
|
|
| namespace MatrixMultiplication.RecursiveCompletion |
|
|
| open MatrixMultiplication.Foundation |
|
|
| inductive Script where |
| | base |
| | step (prior : Script) (center : Color) (size : ℕ) |
|
|
| namespace Script |
|
|
| def factors : Script → ℕ |
| | .base => 1 |
| | .step prior _ m => prior.factors * m |
|
|
| def Coord (U : Type*) : Script → Type _ |
| | .base => U |
| | .step prior _ m => Fin m → Coord U prior |
| |
| instance coordFintype (U : Type*) [Fintype U] (s : Script) : Fintype (Coord U s) := by |
| induction s with |
| | base => exact inferInstanceAs (Fintype U) |
| | step prior center m ih => |
| letI := ih |
| exact inferInstanceAs (Fintype (Fin m → Coord U prior)) |
| |
| def tensor {X Y Z : Type*} (S : FlaggedTensor X Y Z) : |
| (s : Script) → FlaggedTensor (Coord X s) (Coord Y s) (Coord Z s) |
| | .base => S |
| | .step prior center m => complete (tensor S prior) center m |
| |
| def order (d : ℕ) : Script → ℕ |
| | .base => d |
| | .step prior _ m => order d prior * m * 2 + 1 |
| |
| def degree (D : ℕ) : Script → ℕ |
| | .base => D |
| | .step prior _ m => degree D prior * m * 2 + 1 + 1 + 1 |
| |
| def approximation {X Y Z : Type*} [Fintype X] [Fintype Y] [Fintype Z] |
| (S : FlaggedTensor X Y Z) {R d D : ℕ} |
| (A : Tensor.PolynomialApproximation S.tensor R d D) (s : Script) : |
| Tensor.PolynomialApproximation (tensor S s).tensor |
| (R ^ s.factors) (order d s) (degree D s) := by |
| induction s with |
| | base => simpa only [tensor, Coord, factors, order, degree, pow_one] using A |
| | step prior center m ih => |
| have next := (completionDegeneration (tensor S prior) center m).compose (ih.power m) |
| simpa only [tensor, Coord, factors, order, degree, pow_mul] using next |
| |
| theorem rank_power {X Y Z : Type*} [Fintype X] [Fintype Y] [Fintype Z] |
| (S : FlaggedTensor X Y Z) {R d D : ℕ} |
| (A : Tensor.PolynomialApproximation S.tensor R d D) (s : Script) (n : ℕ) : |
| Tensor.RankAtMost (Tensor.power (tensor S s).tensor n) |
| ((degree D s * n + 1) * (R ^ s.factors) ^ n) := |
| (approximation S A s).rank_power n |
| |
| def dual : Script := |
| step (step (step (step (step (step (step base .B 2) .A 2) .C 2) .B 3) .C 2) .B 2) .A 3 |
| |
| def curve : Script := step base .A 3 |
| |
| def square : Script := |
| step (step (step (step (step base .B 2) .A 3) .C 2) .B 2) .A 3 |
| |
| @[simp] theorem dual_factors : dual.factors = 288 := by rfl |
| @[simp] theorem curve_factors : curve.factors = 3 := by rfl |
| @[simp] theorem square_factors : square.factors = 72 := by rfl |
| |
| end Script |
| end MatrixMultiplication.RecursiveCompletion |
| |
| end |
| |
| end OAI |
| |
