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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 | import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.OrbitData
import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.PopulationCWNonzero
import OAI.LinearAlgebra.MatrixMultiplication.CoppersmithWinograd.CWShapePhysicalPermutation
import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.StatisticComplement
namespace OAI
/-! Group assignments, orbit counts and extraction capacities. -/
noncomputable section
namespace MatrixMultiplication.AllFieldGroupCompatibilityTransfer
open MatrixMultiplication.Foundation AllFieldHistory AllFieldHistoryChildLaws
open AllFieldHistoryGroupMasks AllFieldHistoryGroupedRecovery AllFieldGroupOrbitData
open JointPopulation JointCanonicalization JointCanonicalCW CWStrands InheritedMasks
open scoped BigOperators
attribute [local instance] Classical.propDecidable
variable {K tick : ℕ} {sigma : Placement}
theorem physical_coarseMask (allocation : Allocation) (m : ℕ) (sigma : Placement)
(e : Targets (K := K) (tick := tick) allocation m sigma)
(a : Fin 3 → Raw (K := K) (tick := tick) allocation m sigma)
(hcoarse : reorder sigma (ownWord allocation m sigma (a 0),
ownWord allocation m sigma (a 1), ownWord allocation m sigma (a 2)) =
AllFieldGroupOrbitData.coarse allocation m sigma e)
(s : Fin 3) :
coarseMask (Counts allocation m sigma)
(fun h => Fin (activeHalfLength h.val) → Fin 7)
(fun h => Fin (activeHalfLength h.val) → Fin 7)
groupCoarse s e (a s) := by
have ht := reorder_injective sigma hcoarse
intro h i
have he := congrArg (fun t => tripleSide s t ⟨h, i⟩) ht
fin_cases s <;> exact he
def halfShape (right : Bool) (h : ActiveOrder K tick sigma)
(u : JointPopulation.Shape) : Fin 3 → ℕ :=
if right then activeParentShape h.val - shapeNat u else shapeNat u
def halfLaw (right : Bool) (side : Fin 3) (h : ActiveOrder K tick sigma)
(u : JointPopulation.Shape) : Statistic h.val → ℝ :=
if right then rightLaw h.val u side else leftLaw h.val u side
theorem physical_half_product_nonzero (F : Type*) [Field F]
(allocation : Allocation) (m : ℕ) (sigma : Placement)
(e : Targets (K := K) (tick := tick) allocation m sigma)
(a : Fin 3 → Raw (K := K) (tick := tick) allocation m sigma)
(hraw : groupRawSource F allocation m sigma (a 0) (a 1) (a 2) ≠ 0)
(hcoarse : reorder sigma (ownWord allocation m sigma (a 0),
ownWord allocation m sigma (a 1), ownWord allocation m sigma (a 2)) =
AllFieldGroupOrbitData.coarse allocation m sigma e)
(right : Bool) (h : ActiveOrder K tick sigma) (u : JointPopulation.Shape)
(hu : 0 < Counts allocation m sigma h u) :
(∏ i : Class (Counts allocation m sigma) e h u,
shapeTensor (F := F) (Fin (activeHalfLength h.val)) (halfShape right h u)
(if right then (a 0 h i.val).2 else (a 0 h i.val).1)
(if right then (a 1 h i.val).2 else (a 1 h i.val).1)
(if right then (a 2 h i.val).2 else (a 2 h i.val).1)) ≠ 0 := by
have hsupp : ∀ (h : ActiveOrder K tick sigma) (u : JointPopulation.Shape),
0 < Counts allocation m sigma h u →
∀ s, shapeNat u s ≤ activeParentShape h.val s := by
intro h u hu s
exact jointCounts_support_le allocation m h.val.val u hu s
have hsource : sourceTensor (F := F) (Counts (K := K) (tick := tick) allocation m sigma)
(Left (fun h => activeHalfLength h.val)) (Right (fun h => activeHalfLength h.val))
(parentTensor (F := F) (fun h : ActiveOrder K tick sigma => activeHalfLength h.val)
(fun h : ActiveOrder K tick sigma => activeHalfLength h.val)
(fun h : ActiveOrder K tick sigma => activeParentShape h.val))
(a 0) (a 1) (a 2) ≠ 0 := by
exact hraw
have hp := JointPopulationCWNonzero.class_half_products_nonzero
(Counts allocation m sigma) (fun h => activeHalfLength h.val)
(fun h => activeHalfLength h.val) (fun h => activeParentShape h.val)
(fun h => AllFieldHistoryRecovery.halfLength_le_eight h.val) hsupp e
(a 0) (a 1) (a 2) hsource
(physical_coarseMask allocation m sigma e a hcoarse 0)
(physical_coarseMask allocation m sigma e a hcoarse 1)
(physical_coarseMask allocation m sigma e a hcoarse 2) h u hu
cases right
· exact hp.1
· exact hp.2
theorem priority_half_product_nonzero (F : Type*) [Field F]
(allocation : Allocation) (m : ℕ) (sigma : Placement)
(e : Targets (K := K) (tick := tick) allocation m sigma)
(a : Fin 3 → Raw (K := K) (tick := tick) allocation m sigma)
(hraw : groupRawSource F allocation m sigma (a 0) (a 1) (a 2) ≠ 0)
(hcoarse : reorder sigma (ownWord allocation m sigma (a 0),
ownWord allocation m sigma (a 1), ownWord allocation m sigma (a 2)) =
AllFieldGroupOrbitData.coarse allocation m sigma e)
(right : Bool) (h : ActiveOrder K tick sigma) (u : JointPopulation.Shape)
(hu : 0 < Counts allocation m sigma h u) :
(∏ i : Class (Counts allocation m sigma) e h u,
shapeTensor (F := F) (Fin (activeHalfLength h.val)) (halfShape right h u ∘ sigma)
(if right then (a (sigma 0) h i.val).2 else (a (sigma 0) h i.val).1)
(if right then (a (sigma 1) h i.val).2 else (a (sigma 1) h i.val).1)
(if right then (a (sigma 2) h i.val).2 else (a (sigma 2) h i.val).1)) ≠ 0 := by
have he := fun i : Class (Counts allocation m sigma) e h u =>
CWShapePhysicalPermutation.shapeTensor_permute (F := F) sigma (halfShape right h u)
(fun s => if right then (a s h i.val).2 else (a s h i.val).1)
simp_rw [he]
exact physical_half_product_nonzero F allocation m sigma e a hraw hcoarse right h u hu
theorem useful_half_window (allocation : Allocation) (m : ℕ) (ε : ℝ)
(sigma : Placement) (s : Fin 3)
(e : Targets (K := K) (tick := tick) allocation m sigma)
(w : Raw (K := K) (tick := tick) allocation m sigma)
(hw : groupIdealSide allocation m ε s sigma e w)
(right : Bool) (h : ActiveOrder K tick sigma) (u : JointPopulation.Shape)
(hu : 0 < Counts allocation m sigma h u) :
typeWindow (halfLaw right s h u) (AllFieldHistoryMasks.childWidth ε h.val)
(fun i : Class (Counts allocation m sigma) e h u =>
statistic h.val (if right then (w h i.val).2 else (w h i.val).1)) := by
cases right
· exact (hw.2 h u hu).1
· exact (hw.2 h u hu).2
theorem own_coarse_guard (allocation : Allocation) (m : ℕ) (sigma : Placement)
(e : Targets (K := K) (tick := tick) allocation m sigma)
(a : Fin 3 → Raw (K := K) (tick := tick) allocation m sigma)
(hcoarse : reorder sigma (ownWord allocation m sigma (a 0),
ownWord allocation m sigma (a 1), ownWord allocation m sigma (a 2)) =
AllFieldGroupOrbitData.coarse allocation m sigma e)
(side : Fin 3) :
tripleSide side (AllFieldGroupOrbitData.coarse allocation m sigma e) =
ownWord allocation m sigma (a (sigma side)) := by
rw [← hcoarse, tripleSide_reorder]
generalize sigma side = s
fin_cases s <;> rfl
theorem transferY (F : Type*) [Field F] (allocation : Allocation) (m : ℕ) (ε : ℝ)
(sigma : Placement) (e : Targets (K := K) (tick := tick) allocation m sigma)
(a : Fin 3 → Raw (K := K) (tick := tick) allocation m sigma)
(hraw : groupRawSource F allocation m sigma (a 0) (a 1) (a 2) ≠ 0)
(hcoarse : reorder sigma (ownWord allocation m sigma (a 0),
ownWord allocation m sigma (a 1), ownWord allocation m sigma (a 2)) =
AllFieldGroupOrbitData.coarse allocation m sigma e)
(huseX : groupIdealSide allocation m ε (sigma 0) sigma e (a (sigma 0))) :
Compatible allocation m ε sigma 1 e (a (sigma 1)) := by
refine ⟨own_coarse_guard allocation m sigma e a hcoarse 1, ?_⟩
intro right h u hd
apply JointPopulationCompatibility.shape_count_window_of_typeWindow
(Counts allocation m sigma) e h u _ _ _
intro hu
have hz : halfShape right h u (sigma 2) = 0 := by
cases right <;> simpa [designated, halfShape, shapeNat] using hd.2
have hw := CWStrandCompatibility.zeroZ_transfer
(fun _ : Class (Counts allocation m sigma) e h u => halfShape right h u ∘ sigma)
(fun i => if right then (a (sigma 0) h i.val).2 else (a (sigma 0) h i.val).1)
(fun i => if right then (a (sigma 1) h i.val).2 else (a (sigma 1) h i.val).1)
(fun i => if right then (a (sigma 2) h i.val).2 else (a (sigma 2) h i.val).1)
(priority_half_product_nonzero F allocation m sigma e a hraw hcoarse right h u hu)
(statistic h.val) (complement h)
(AllFieldGroupStatisticComplement.statistic_wordComplement h)
(halfLaw right (sigma 0) h u) (AllFieldHistoryMasks.childWidth ε h.val)
(fun _ => hz) (useful_half_window allocation m ε sigma (sigma 0) e _ huseX right h u hu)
simpa [compatibilityLaw, halfLaw] using hw
theorem transferZ (F : Type*) [Field F] (allocation : Allocation) (m : ℕ) (ε : ℝ)
(sigma : Placement) (e : Targets (K := K) (tick := tick) allocation m sigma)
(a : Fin 3 → Raw (K := K) (tick := tick) allocation m sigma)
(hraw : groupRawSource F allocation m sigma (a 0) (a 1) (a 2) ≠ 0)
(hcoarse : reorder sigma (ownWord allocation m sigma (a 0),
ownWord allocation m sigma (a 1), ownWord allocation m sigma (a 2)) =
AllFieldGroupOrbitData.coarse allocation m sigma e)
(huseX : groupIdealSide allocation m ε (sigma 0) sigma e (a (sigma 0)))
(huseY : groupIdealSide allocation m ε (sigma 1) sigma e (a (sigma 1))) :
Compatible allocation m ε sigma 2 e (a (sigma 2)) := by
refine ⟨own_coarse_guard allocation m sigma e a hcoarse 2, ?_⟩
intro right h u hd
apply JointPopulationCompatibility.shape_count_window_of_typeWindow
(Counts allocation m sigma) e h u _ _ _
intro hu
have hz : halfShape right h u (sigma 0) * halfShape right h u (sigma 1) = 0 := by
cases right <;> simpa [designated, halfShape, shapeNat] using hd.2
have hp := priority_half_product_nonzero F allocation m sigma e a hraw hcoarse right h u hu
by_cases hy : halfShape right h u (sigma 1) = 0
· have hw := CWStrandCompatibility.zeroY_transfer
(fun _ : Class (Counts allocation m sigma) e h u => halfShape right h u ∘ sigma)
(fun i => if right then (a (sigma 0) h i.val).2 else (a (sigma 0) h i.val).1)
(fun i => if right then (a (sigma 1) h i.val).2 else (a (sigma 1) h i.val).1)
(fun i => if right then (a (sigma 2) h i.val).2 else (a (sigma 2) h i.val).1)
hp (statistic h.val) (complement h)
(AllFieldGroupStatisticComplement.statistic_wordComplement h)
(halfLaw right (sigma 0) h u) (AllFieldHistoryMasks.childWidth ε h.val)
(fun _ => hy) (useful_half_window allocation m ε sigma (sigma 0) e _ huseX right h u hu)
have hy' : (if right then activeParentShape h.val (sigma 1) -
(shapeSide (sigma 1) u).val else (shapeSide (sigma 1) u).val) = 0 := by
cases right <;> simpa [halfShape, shapeNat] using hy
simpa [compatibilityLaw, halfLaw, hy'] using hw
· have hx : halfShape right h u (sigma 0) = 0 := (Nat.mul_eq_zero.mp hz).resolve_right hy
have hw := CWStrandCompatibility.zeroX_transfer
(fun _ : Class (Counts allocation m sigma) e h u => halfShape right h u ∘ sigma)
(fun i => if right then (a (sigma 0) h i.val).2 else (a (sigma 0) h i.val).1)
(fun i => if right then (a (sigma 1) h i.val).2 else (a (sigma 1) h i.val).1)
(fun i => if right then (a (sigma 2) h i.val).2 else (a (sigma 2) h i.val).1)
hp (statistic h.val) (complement h)
(AllFieldGroupStatisticComplement.statistic_wordComplement h)
(halfLaw right (sigma 1) h u) (AllFieldHistoryMasks.childWidth ε h.val)
(fun _ => hx) (useful_half_window allocation m ε sigma (sigma 1) e _ huseY right h u hu)
have hy' : (if right then activeParentShape h.val (sigma 1) -
(shapeSide (sigma 1) u).val else (shapeSide (sigma 1) u).val) ≠ 0 := by
cases right <;> simpa [halfShape, shapeNat] using hy
simpa [compatibilityLaw, halfLaw, hy'] using hw
end MatrixMultiplication.AllFieldGroupCompatibilityTransfer
end
end OAI
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