File size: 2,691 Bytes
cdec422 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 | 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
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