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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 | import OAI.LinearAlgebra.MatrixMultiplication.CoppersmithWinograd.CWZeroGeometry
import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.Tensors
import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.InitialLeafRates
import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.Terminal
namespace OAI
/-! Tensor extraction over arbitrary fields and its asymptotic rate. -/
noncomputable section
namespace MatrixMultiplication.AllFieldInitialZeroFinish
open MatrixMultiplication.Foundation AllFieldParameters AllFieldHistory AllFieldFiniteFamily
open AllFieldInitialLeafRates CWWindowedLeaves CWStrands CWZeroGeometry
open scoped BigOperators Classical
variable {K : ℕ}
abbrev Index (K : ℕ) := PopulationHistory K
local instance : DecidableEq (Index K) := Classical.decEq _
theorem shape_total (h : InitialZero K) : shapeTotal (initialShape h.val) = 16 :=
(root_shape_spec _ (initialShape_mem h.val)).2
theorem has_zero (h : InitialZero K) : ∃ i, initialShape h.val i = 0 :=
terminal_has_zero (.initial h.val) h.property
theorem canonical_axes_ne (h : InitialZero K) :
zeroAxis (initialShape h.val) ≠ maxAxis (initialShape h.val) :=
zero_ne_selected _ _ _ (by rw [shape_total]; norm_num)
(zeroAxis_spec _ (has_zero h)) (maxAxis_spec _)
def zero (h : Index K) : Fin 3 := h.2 (zeroAxis (initialShape h.1.val))
def selected (h : Index K) : Fin 3 := h.2 (maxAxis (initialShape h.1.val))
theorem axes_ne (h : Index K) : zero h ≠ selected h :=
fun he => canonical_axes_ne h.1 (h.2.injective he)
theorem weights_sum (h : InitialZero K) :
shapeMax (initialShape h.val) + (16 - shapeMax (initialShape h.val)) = 2 * 8 := by
simpa only [shape_total h] using max_add_complement (initialShape h.val)
theorem physical_shape_eq (h : Index K) :
physicalShape h.2 (initialShape h.1.val) =
CWOrientedZero.placedShape (zero h) (selected h)
(shapeMax (initialShape h.1.val)) (16 - shapeMax (initialShape h.1.val)) := by
simpa only [zero, selected, shape_total h.1] using
physicalShape_eq_placedShape (initialShape h.1.val) h.2
(zeroAxis (initialShape h.1.val)) (maxAxis (initialShape h.1.val))
(zeroAxis_spec _ (has_zero h.1)) (maxAxis_spec _) (canonical_axes_ne h.1)
variable (allocation : Allocation) (dilation : ℕ)
def embed (h : Index K) (side : Fin 3) (w : Words allocation dilation h) :
HistoryWord allocation dilation (.initial h.1.val, h.2) :=
fun i => CWOrientedZero.sideWord (zero h) (selected h) side (w i)
theorem coefficient (F : Type*) [Field F] (ε : ℝ) (h : Index K)
(x y z : Words allocation dilation h) :
historyTensor F allocation dilation ε (.initial h.1.val, h.2)
(embed allocation dilation h 0 x) (embed allocation dilation h 1 y)
(embed allocation dilation h 2 z) =
if CWOrientedZero.matched (zero h) x y z then 1 else 0 := by
rw [historyTensor_initial, physical_shape_eq]
exact CWOrientedZero.power_coefficient F 8 (shapeMax (initialShape h.1.val))
(16 - shapeMax (initialShape h.1.val))
(population allocation dilation (.initial h.1.val, h.2))
(weights_sum h.1) (zero h) (selected h) (axes_ne h)
(fun w : Words allocation dilation h => w) Function.injective_id x y z
abbrev RowIndex (h : Index K) := CWOrientedZero.Row (zero h) (Words allocation dilation h)
abbrev MiddleIndex (h : Index K) := CWOrientedZero.Middle (zero h) (Words allocation dilation h)
abbrev ColumnIndex (h : Index K) := CWOrientedZero.Column (zero h) (Words allocation dilation h)
def factorMap (F : Type*) [Field F] (ε : ℝ) (h : Index K) :
LocalMap (historyTensor F allocation dilation ε (.initial h.1.val, h.2))
(Tensor.matrixCoefficients (RowIndex allocation dilation h)
(MiddleIndex allocation dilation h) (ColumnIndex allocation dilation h)) := by
letI : Nonempty (Words allocation dilation h) :=
Fintype.card_pos_iff.mp (card_words_pos allocation dilation h)
exact CWOrientedZero.ofMatching F _ (zero h) (embed allocation dilation h)
(coefficient allocation dilation F ε h)
theorem factor_volume (h : Index K) :
Fintype.card (RowIndex allocation dilation h) *
Fintype.card (MiddleIndex allocation dilation h) *
Fintype.card (ColumnIndex allocation dilation h) =
Fintype.card (Words allocation dilation h) :=
CWOrientedZero.volume (zero h) (Words allocation dilation h)
abbrev RawWords := ∀ h : Index K,
HistoryWord allocation dilation (.initial h.1.val, h.2)
abbrev Rows := ∀ h : Index K, RowIndex allocation dilation h
abbrev Middles := ∀ h : Index K, MiddleIndex allocation dilation h
abbrev Columns := ∀ h : Index K, ColumnIndex allocation dilation h
def source (F : Type*) [Field F] (ε : ℝ) :
Tensor F (RawWords (K := K) allocation dilation)
(RawWords (K := K) allocation dilation) (RawWords (K := K) allocation dilation) :=
CommonDimensions.familyProduct (fun h : Index K =>
historyTensor F allocation dilation ε (.initial h.1.val, h.2))
def finishMap (F : Type*) [Field F] (ε : ℝ) :
LocalMap (source (K := K) allocation dilation F ε)
(Tensor.matrixCoefficients (Rows (K := K) allocation dilation)
(Middles (K := K) allocation dilation) (Columns (K := K) allocation dilation)) :=
AllFieldTerminal.terminalProductMap
(H := Index K)
(X := fun h => HistoryWord allocation dilation (.initial h.1.val, h.2))
(Y := fun h => HistoryWord allocation dilation (.initial h.1.val, h.2))
(Z := fun h => HistoryWord allocation dilation (.initial h.1.val, h.2))
(fun h : Index K => historyTensor F allocation dilation ε (.initial h.1.val, h.2))
(RowIndex allocation dilation) (MiddleIndex allocation dilation)
(ColumnIndex allocation dilation) (factorMap allocation dilation F ε)
theorem volume_eq :
Fintype.card (Rows (K := K) allocation dilation) *
Fintype.card (Middles (K := K) allocation dilation) *
Fintype.card (Columns (K := K) allocation dilation) =
AllFieldInitialLeafRates.volume (K := K) allocation dilation := by
rw [AllFieldInitialLeafRates.volume_eq_prod]
simp only [Rows, Middles, Columns, Fintype.card_pi,
← Finset.prod_mul_distrib]
apply Finset.prod_congr rfl
intro h _
simpa only [Words, Fintype.card_pi] using factor_volume allocation dilation h
theorem volume_pos :
0 < Fintype.card (Rows (K := K) allocation dilation) *
Fintype.card (Middles (K := K) allocation dilation) *
Fintype.card (Columns (K := K) allocation dilation) := by
rw [volume_eq]
exact AllFieldInitialLeafRates.volume_pos allocation dilation
theorem rows_pos : 0 < Fintype.card (Rows (K := K) allocation dilation) :=
Nat.pos_of_mul_pos_right (Nat.pos_of_mul_pos_right (volume_pos allocation dilation))
theorem middles_pos : 0 < Fintype.card (Middles (K := K) allocation dilation) :=
Nat.pos_of_mul_pos_left (Nat.pos_of_mul_pos_right (volume_pos allocation dilation))
theorem columns_pos : 0 < Fintype.card (Columns (K := K) allocation dilation) :=
Nat.pos_of_mul_pos_left (volume_pos allocation dilation)
end MatrixMultiplication.AllFieldInitialZeroFinish
end
end OAI
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