File size: 3,583 Bytes
f18e7de | 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 | import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.CanonicalCW
namespace OAI
/-! Joint tensor extraction, compatibility and entropy estimates. -/
noncomputable section
namespace MatrixMultiplication.JointPopulationCWNonzero
open MatrixMultiplication.Foundation JointPopulation JointCanonicalization JointCanonicalCW
open CWStrands InheritedMasks
open scoped BigOperators
attribute [local instance] Classical.propDecidable
variable {H F : Type*} [Fintype H] [DecidableEq H] [Field F]
omit [DecidableEq H] in
theorem class_half_products_nonzero
(counts : H → Shape → ℕ) (leftLength rightLength : H → ℕ)
(parentShape : H → Fin 3 → ℕ) (hleft : ∀ h, leftLength h ≤ 8)
(hsupp : ∀ h u, 0 < counts h u → ∀ s, shapeNat u s ≤ parentShape h s)
(e : Target counts)
(x y z : RawPairs counts (Left leftLength) (Right rightLength))
(hsource : sourceTensor counts (Left leftLength) (Right rightLength)
(parentTensor (F := F) leftLength rightLength parentShape) x y z ≠ 0)
(hx : coarseMask counts (Left leftLength) (Right rightLength)
(coarse leftLength rightLength hleft) 0 e x)
(hy : coarseMask counts (Left leftLength) (Right rightLength)
(coarse leftLength rightLength hleft) 1 e y)
(hz : coarseMask counts (Left leftLength) (Right rightLength)
(coarse leftLength rightLength hleft) 2 e z)
(h : H) (u : Shape) (hn : 0 < counts h u) :
(∏ i : Class counts e h u,
shapeTensor (F := F) (Fin (leftLength h)) (shapeNat u)
(x h i.val).1 (y h i.val).1 (z h i.val).1) ≠ 0 ∧
(∏ i : Class counts e h u,
shapeTensor (F := F) (Fin (rightLength h)) (parentShape h - shapeNat u)
(x h i.val).2 (y h i.val).2 (z h i.val).2) ≠ 0 := by
change (∏ h' : H, ∏ i : Positions counts h',
parentTensor (F := F) leftLength rightLength parentShape h'
(x h' i) (y h' i) (z h' i)) ≠ 0 at hsource
have hparent (i : Positions counts h) :
parentTensor (F := F) leftLength rightLength parentShape h
(x h i) (y h i) (z h i) ≠ 0 :=
Finset.prod_ne_zero_iff.mp
(Finset.prod_ne_zero_iff.mp hsource h (Finset.mem_univ h)) i (Finset.mem_univ i)
have hhalves (i : Class counts e h u) :
shapeTensor (F := F) (Fin (leftLength h)) (shapeNat u)
(x h i.val).1 (y h i.val).1 (z h i.val).1 *
shapeTensor (F := F) (Fin (rightLength h)) (parentShape h - shapeNat u)
(x h i.val).2 (y h i.val).2 (z h i.val).2 ≠ 0 := by
have hx' : coarse leftLength rightLength hleft 0 h (x h i.val) = u.1 := by
simpa [shapeSide, i.property] using hx h i.val
have hy' : coarse leftLength rightLength hleft 1 h (y h i.val) = u.2.1 := by
simpa [shapeSide, i.property] using hy h i.val
have hz' : coarse leftLength rightLength hleft 2 h (z h i.val) = u.2.2 := by
simpa [shapeSide, i.property] using hz h i.val
have hblock : block (Left leftLength) (Right rightLength)
(parentTensor (F := F) leftLength rightLength parentShape)
(coarse leftLength rightLength hleft) (h, u)
(x h i.val) (y h i.val) (z h i.val) ≠ 0 := by
simpa [block, hx', hy', hz'] using hparent i.val
rw [block_eq_product leftLength rightLength parentShape hleft h u
(hsupp h u hn)] at hblock
exact hblock
constructor
· apply Finset.prod_ne_zero_iff.mpr
intro i _
exact (mul_ne_zero_iff.mp (hhalves i)).1
· apply Finset.prod_ne_zero_iff.mpr
intro i _
exact (mul_ne_zero_iff.mp (hhalves i)).2
end MatrixMultiplication.JointPopulationCWNonzero
end
end OAI
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