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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 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 | import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.Canonicalization
import OAI.LinearAlgebra.MatrixMultiplication.CoppersmithWinograd.CWCoarseIndices
namespace OAI
/-! Joint tensor extraction, compatibility and entropy estimates. -/
noncomputable section
namespace MatrixMultiplication.JointCanonicalCW
open MatrixMultiplication.Foundation JointPopulation JointCanonicalization
open CWStrands HistorySymmetry InheritedMasks
open scoped BigOperators
attribute [local instance] Classical.propDecidable
variable {H F : Type*} [Fintype H] [DecidableEq H] [CommRing F]
(counts : H → Shape → ℕ) (leftLength rightLength : H → ℕ)
(parentShape : H → Fin 3 → ℕ)
(hleft : ∀ h, leftLength h ≤ 8)
abbrev Left (h : H) := Fin (leftLength h) → Fin 7
abbrev Right (h : H) := Fin (rightLength h) → Fin 7
def shapeNat (u : Shape) : Fin 3 → ℕ := fun s => (shapeSide s u).val
def parentTensor (h : H) : Tensor F
(Left leftLength h × Right rightLength h)
(Left leftLength h × Right rightLength h)
(Left leftLength h × Right rightLength h) :=
fun x y z => shapeTensor (Fin (leftLength h) ⊕ Fin (rightLength h)) (parentShape h)
(Sum.elim x.1 x.2) (Sum.elim y.1 y.2) (Sum.elim z.1 z.2)
def coarse (_s : Fin 3) (h : H) (w : Left leftLength h × Right rightLength h) : Fin 17 :=
CWCoarseIndices.coarseIndex (by simpa using hleft h) w.1
def leftTensor (c : ClassKey (H := H)) :
Tensor F (Left leftLength c.1) (Left leftLength c.1) (Left leftLength c.1) :=
shapeTensor (Fin (leftLength c.1)) (shapeNat c.2)
def rightTensor (c : ClassKey (H := H)) :
Tensor F (Right rightLength c.1) (Right rightLength c.1) (Right rightLength c.1) :=
shapeTensor (Fin (rightLength c.1)) (parentShape c.1 - shapeNat c.2)
omit [Fintype H] [DecidableEq H] in
theorem block_eq_product (h : H) (u : Shape)
(hu : ∀ s, shapeNat u s ≤ parentShape h s) :
block (Left leftLength) (Right rightLength)
(parentTensor leftLength rightLength parentShape)
(coarse leftLength rightLength hleft) (h, u) =
Tensor.product (leftTensor (F := F) leftLength (h, u))
(rightTensor rightLength parentShape (h, u)) := by
have hadd : shapeNat u + (parentShape h - shapeNat u) = parentShape h := by
funext s
exact Nat.add_sub_of_le (hu s)
funext x y z
have hs := shape_split (F := F) (P := Fin (leftLength h))
(Q := Fin (rightLength h)) (shapeNat u) (parentShape h - shapeNat u) x y z
rw [hadd] at hs
simpa [block, parentTensor, coarse, CWCoarseIndices.coarseIndex,
Fin.ext_iff, leftTensor, rightTensor, shapeNat, shapeSide,
Fin.isValue, ↓reduceIte] using hs
omit [DecidableEq H] in
theorem canonicalBase_eq_children
(hsupport : ∀ h u, 0 < counts h u → ∀ s, shapeNat u s ≤ parentShape h s) :
canonicalBase counts (Left leftLength) (Right rightLength)
(parentTensor leftLength rightLength parentShape)
(coarse leftLength rightLength hleft) =
pairClassProduct (leftTensor (F := F) leftLength)
(rightTensor rightLength parentShape) := by
funext x y z
unfold canonicalBase
have hf : (∏ c : ClassKey (H := H), ∏ j : ClassPositions counts c,
block (Left leftLength) (Right rightLength)
(parentTensor leftLength rightLength parentShape)
(coarse leftLength rightLength hleft) c
(x.left c j, x.right c j) (y.left c j, y.right c j) (z.left c j, z.right c j)) =
∏ c : ClassKey (H := H), ∏ j : ClassPositions counts c,
(leftTensor (F := F) leftLength c (x.left c j) (y.left c j) (z.left c j) *
rightTensor rightLength parentShape c (x.right c j) (y.right c j) (z.right c j)) := by
apply Finset.prod_congr rfl
intro c _
apply Finset.prod_congr rfl
intro j _
have hc : 0 < counts c.1 c.2 := lt_of_le_of_lt (Nat.zero_le j.val) j.isLt
rw [block_eq_product leftLength rightLength parentShape hleft c.1 c.2
(hsupport c.1 c.2 hc)]
rfl
rw [hf]
simp only [pairClassProduct, classProduct, Finset.prod_mul_distrib]
abbrev RawStrands := ∀ h, Positions counts h →
((Fin (leftLength h) ⊕ Fin (rightLength h)) → Fin 7)
def rawParentSource : Tensor F (RawStrands counts leftLength rightLength)
(RawStrands counts leftLength rightLength) (RawStrands counts leftLength rightLength) :=
classProduct (fun h => shapeTensor (Fin (leftLength h) ⊕ Fin (rightLength h)) (parentShape h))
def joinWords (w : RawPairs counts (Left leftLength) (Right rightLength)) :
RawStrands counts leftLength rightLength :=
fun h i => Sum.elim (w h i).1 (w h i).2
def joinMatrix : RawPairs counts (Left leftLength) (Right rightLength) →
RawStrands counts leftLength rightLength → F :=
fun w v => if v = joinWords counts leftLength rightLength w then 1 else 0
theorem parentTensor_restrict :
Tensor.restrict (joinMatrix (F := F) counts leftLength rightLength)
(joinMatrix counts leftLength rightLength) (joinMatrix counts leftLength rightLength)
(rawParentSource counts leftLength rightLength parentShape) =
sourceTensor counts (Left leftLength) (Right rightLength)
(parentTensor leftLength rightLength parentShape) := by
classical
funext x y z
simp [Tensor.restrict, joinMatrix, rawParentSource, sourceTensor,
classProduct, joinWords, parentTensor, ite_mul, mul_ite]
end MatrixMultiplication.JointCanonicalCW
end
end OAI
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