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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 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 | import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.PopulationCompatibility
import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.PopulationMixedNonzero
import OAI.LinearAlgebra.MatrixMultiplication.CoppersmithWinograd.CWStrandCompatibility
namespace OAI
/-! Joint tensor extraction, compatibility and entropy estimates. -/
noncomputable section
namespace MatrixMultiplication.JointPopulationMixedCompatibility
open MatrixMultiplication.Foundation JointPopulation JointCanonicalization JointCanonicalCW
open JointPopulationCompatibility JointPopulationMixedNonzero CWStrands InheritedMasks
open CWStrandCompatibility
open scoped BigOperators
attribute [local instance] Classical.propDecidable
def sharingDesignated {H : Type*} (sharing : H → Bool)
(d : JointPopulationCompatibility.ClassKey (H := H) → Shape → Prop)
(c : JointPopulationCompatibility.ClassKey (H := H)) (u : Shape) : Prop :=
sharing c.1 = true ∧ d c u
variable {H F : Type*} [Fintype H] [DecidableEq H] [Field F]
(counts : H → Shape → ℕ) (leftLength rightLength : H → ℕ)
(parentShape : H → Fin 3 → ℕ) (sharing : H → Bool) (hleft : ∀ h, leftLength h ≤ 8)
(hsupport : ∀ h u, 0 < counts h u → sharing h = true → ∀ 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)
(JointCanonicalMixed.inputTensor (F := F) leftLength rightLength parentShape sharing) 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)
variable (SL SR : H → Type*)
[∀ h, Fintype (SL h)] [∀ h, DecidableEq (SL h)]
[∀ h, Fintype (SR h)] [∀ h, DecidableEq (SR h)]
(leftStatistic : ∀ h, Left leftLength h → SL h)
(rightStatistic : ∀ h, Right rightLength h → SR h)
(κLeft : ∀ h, SL h ≃ SL h) (κRight : ∀ h, SR h ≃ SR h)
(hstatLeft : ∀ h w, leftStatistic h (CWCompatibilityTransfer.wordComplement w) =
κLeft h (leftStatistic h w))
(hstatRight : ∀ h w, rightStatistic h (CWCompatibilityTransfer.wordComplement w) =
κRight h (rightStatistic h w))
(xLeftLaw yLeftLaw : ∀ c : JointCanonicalization.ClassKey (H := H), SL c.1 → ℝ)
(xRightLaw yRightLaw : ∀ c : JointCanonicalization.ClassKey (H := H), SR c.1 → ℝ)
(χ : ℝ)
include hsource hx hy hz hsupport hstatLeft hstatRight
omit [DecidableEq H] in
theorem source_implies_compatibleY
(leftLaw : ∀ c : JointPopulationCompatibility.ClassKey (H := H), Shape → SL c.1 → ℝ)
(rightLaw : ∀ c : JointPopulationCompatibility.ClassKey (H := H), Shape → SR c.1 → ℝ)
(hlaw : ∀ h u, 0 < counts h u → sharing h = true → designatedYLeft (h, shapeSide 1 u) u →
leftLaw (h, shapeSide 1 u) u = xLeftLaw (h, u) ∘ (κLeft h).symm)
(hrlaw : ∀ h u, 0 < counts h u → sharing h = true → designatedYRight parentShape (h, shapeSide 1 u) u →
rightLaw (h, shapeSide 1 u) u = xRightLaw (h, u) ∘ (κRight h).symm)
(huseX : ∀ h u, 0 < counts h u → sharing h = true →
typeWindow (xLeftLaw (h, u)) χ
(fun i : Class counts e h u => leftStatistic h (x h i.val).1) ∧
typeWindow (xRightLaw (h, u)) χ
(fun i : Class counts e h u => rightStatistic h (x h i.val).2)) :
Compatible counts (Left leftLength) (Right rightLength)
(fun c => SL c.1) (fun c => SR c.1)
(fun c => leftStatistic c.1) (fun c => rightStatistic c.1)
(sharingDesignated sharing designatedYLeft)
(sharingDesignated sharing (designatedYRight parentShape)) leftLaw rightLaw χ 1 e y := by
apply compatible_of_designated_windows
· intro h u hn hd
rw [hlaw h u hn hd.1 hd.2]
exact zeroZ_transfer (fun _ : Class counts e h u => shapeNat u)
(fun i => (x h i.val).1) (fun i => (y h i.val).1) (fun i => (z h i.val).1)
(class_half_products_nonzero counts leftLength rightLength parentShape sharing hleft
hsupport e x y z hsource hx hy hz h u hn hd.1).1
(leftStatistic h) (κLeft h) (hstatLeft h) (xLeftLaw (h, u)) χ
(fun _ => hd.2) (huseX h u hn hd.1).1
· intro h u hn hd
rw [hrlaw h u hn hd.1 hd.2]
exact zeroZ_transfer (fun _ : Class counts e h u => parentShape h - shapeNat u)
(fun i => (x h i.val).2) (fun i => (y h i.val).2) (fun i => (z h i.val).2)
(class_half_products_nonzero counts leftLength rightLength parentShape sharing hleft
hsupport e x y z hsource hx hy hz h u hn hd.1).2
(rightStatistic h) (κRight h) (hstatRight h) (xRightLaw (h, u)) χ
(fun _ => hd.2) (huseX h u hn hd.1).2
omit [DecidableEq H] in
theorem source_implies_compatibleZ
(leftLaw : ∀ c : JointPopulationCompatibility.ClassKey (H := H), Shape → SL c.1 → ℝ)
(rightLaw : ∀ c : JointPopulationCompatibility.ClassKey (H := H), Shape → SR c.1 → ℝ)
(hlaw : ∀ h u, 0 < counts h u → sharing h = true → designatedZLeft (h, shapeSide 2 u) u →
leftLaw (h, shapeSide 2 u) u =
(if shapeNat u 1 = 0 then xLeftLaw (h, u) else yLeftLaw (h, u)) ∘
(κLeft h).symm)
(hrlaw : ∀ h u, 0 < counts h u → sharing h = true → designatedZRight parentShape (h, shapeSide 2 u) u →
rightLaw (h, shapeSide 2 u) u =
(if (parentShape h - shapeNat u) 1 = 0 then xRightLaw (h, u)
else yRightLaw (h, u)) ∘ (κRight h).symm)
(huseX : ∀ h u, 0 < counts h u → sharing h = true →
typeWindow (xLeftLaw (h, u)) χ
(fun i : Class counts e h u => leftStatistic h (x h i.val).1) ∧
typeWindow (xRightLaw (h, u)) χ
(fun i : Class counts e h u => rightStatistic h (x h i.val).2))
(huseY : ∀ h u, 0 < counts h u → sharing h = true →
typeWindow (yLeftLaw (h, u)) χ
(fun i : Class counts e h u => leftStatistic h (y h i.val).1) ∧
typeWindow (yRightLaw (h, u)) χ
(fun i : Class counts e h u => rightStatistic h (y h i.val).2)) :
Compatible counts (Left leftLength) (Right rightLength)
(fun c => SL c.1) (fun c => SR c.1)
(fun c => leftStatistic c.1) (fun c => rightStatistic c.1)
(sharingDesignated sharing designatedZLeft)
(sharingDesignated sharing (designatedZRight parentShape)) leftLaw rightLaw χ 2 e z := by
apply compatible_of_designated_windows
· intro h u hn hd
rw [hlaw h u hn hd.1 hd.2]
have hp := (class_half_products_nonzero counts leftLength rightLength parentShape sharing hleft
hsupport e x y z hsource hx hy hz h u hn hd.1).1
by_cases hzeroY : shapeNat u 1 = 0
· rw [ite_eq_left hzeroY]
exact zeroY_transfer (fun _ : Class counts e h u => shapeNat u)
(fun i => (x h i.val).1) (fun i => (y h i.val).1) (fun i => (z h i.val).1) hp
(leftStatistic h) (κLeft h) (hstatLeft h) (xLeftLaw (h, u)) χ
(fun _ => hzeroY) (huseX h u hn hd.1).1
· rw [ite_eq_right hzeroY]
have hzeroX : shapeNat u 0 = 0 :=
(Nat.mul_eq_zero.mp hd.2).resolve_right hzeroY
exact zeroX_transfer (fun _ : Class counts e h u => shapeNat u)
(fun i => (x h i.val).1) (fun i => (y h i.val).1) (fun i => (z h i.val).1) hp
(leftStatistic h) (κLeft h) (hstatLeft h) (yLeftLaw (h, u)) χ
(fun _ => hzeroX) (huseY h u hn hd.1).1
· intro h u hn hd
rw [hrlaw h u hn hd.1 hd.2]
have hp := (class_half_products_nonzero counts leftLength rightLength parentShape sharing hleft
hsupport e x y z hsource hx hy hz h u hn hd.1).2
by_cases hzeroY : (parentShape h - shapeNat u) 1 = 0
· rw [ite_eq_left hzeroY]
exact zeroY_transfer (fun _ : Class counts e h u => parentShape h - shapeNat u)
(fun i => (x h i.val).2) (fun i => (y h i.val).2) (fun i => (z h i.val).2) hp
(rightStatistic h) (κRight h) (hstatRight h) (xRightLaw (h, u)) χ
(fun _ => hzeroY) (huseX h u hn hd.1).2
· rw [ite_eq_right hzeroY]
have hzeroX : (parentShape h - shapeNat u) 0 = 0 :=
(Nat.mul_eq_zero.mp hd.2).resolve_right hzeroY
exact zeroX_transfer (fun _ : Class counts e h u => parentShape h - shapeNat u)
(fun i => (x h i.val).2) (fun i => (y h i.val).2) (fun i => (z h i.val).2) hp
(rightStatistic h) (κRight h) (hstatRight h) (yRightLaw (h, u)) χ
(fun _ => hzeroX) (huseY h u hn hd.1).2
end MatrixMultiplication.JointPopulationMixedCompatibility
end
end OAI
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