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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 | import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.ProducedBase
import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.ProducedMasks
import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.RecoveryCore
import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.Regrouping
namespace OAI
/-! Finite extraction histories, inherited masks and recovery bounds. -/
noncomputable section
namespace MatrixMultiplication.AllFieldHistoryProduced
open MatrixMultiplication.Foundation AllFieldHistory AllFieldFiniteFamily
open AllFieldHistoryProducedMasks
open scoped BigOperators Classical
attribute [local instance] Classical.propDecidable Classical.decEq
variable {K tick : ℕ} (allocation : Allocation) (m : ℕ)
def passes (ε : ℝ) (side : Fin 3)
(w : AllFieldHistory.ProducedWords (K := K) (tick := tick) allocation m) : Prop :=
∀ h : Produced K tick, residentMask allocation m ε h.val side (w h)
theorem producedTensor_eq_delete (F : Type*) [Field F] (ε : ℝ) :
AllFieldHistoryRegrouping.producedTensor (K := K) (tick := tick) allocation m F ε =
ExactRecovery.delete (producedBase (F := F) (K := K) (tick := tick) allocation m)
(passes allocation m ε 0) (passes allocation m ε 1) (passes allocation m ε 2) := by
funext x y z
simp only [AllFieldHistoryRegrouping.producedTensor, CommonDimensions.familyProduct,
historyTensor, ExactRecovery.delete, Fintype.prod_ite_zero,
passes, forall_and, producedBase]
theorem canonical_eq_delete (F : Type*) [Field F] (ε : ℝ) :
AllFieldHistoryRecovery.canonical F (K := K) (tick := tick) allocation m ε =
ExactRecovery.delete (canonicalChildren (F := F) (K := K) (tick := tick) allocation m)
(AllFieldHistoryMasks.childWindows allocation m ε 0)
(AllFieldHistoryMasks.childWindows allocation m ε 1)
(AllFieldHistoryMasks.childWindows allocation m ε 2) := by
have hb := JointCanonicalCW.canonicalBase_eq_children (F := F)
(activeCounts (K := K) (tick := tick) allocation m)
activeHalfLength activeHalfLength activeParentShape
AllFieldHistoryRecovery.halfLength_le_eight
(fun h u hu => jointCounts_support_le allocation m h.val u hu)
unfold AllFieldHistoryRecovery.canonical JointCanonicalization.canonicalIdeal
change ExactRecovery.delete
(JointCanonicalization.canonicalBase (activeCounts allocation m)
(JointCanonicalCW.Left activeHalfLength) (JointCanonicalCW.Right activeHalfLength)
(JointCanonicalCW.parentTensor activeHalfLength activeHalfLength activeParentShape)
(JointCanonicalCW.coarse activeHalfLength activeHalfLength
AllFieldHistoryRecovery.halfLength_le_eight))
(AllFieldHistoryMasks.childWindows allocation m ε 0)
(AllFieldHistoryMasks.childWindows allocation m ε 1)
(AllFieldHistoryMasks.childWindows allocation m ε 2) = _
rw [hb]
rfl
def pulled (F : Type*) [Field F] (ε : ℝ) :
Tensor F (AllFieldHistory.ProducedWords (K := K) (tick := tick) allocation m)
(AllFieldHistory.ProducedWords (K := K) (tick := tick) allocation m)
(AllFieldHistory.ProducedWords (K := K) (tick := tick) allocation m) :=
Tensor.pullback (regroupWords allocation m) (regroupWords allocation m)
(regroupWords allocation m)
(AllFieldHistoryRecovery.canonical F (K := K) (tick := tick) allocation m ε)
theorem delete_pulled (F : Type*) [Field F] {ε : ℝ} (hε : 0 ≤ ε) :
ExactRecovery.delete (pulled (K := K) (tick := tick) allocation m F ε)
(passes allocation m ε 0) (passes allocation m ε 1) (passes allocation m ε 2) =
AllFieldHistoryRegrouping.producedTensor (K := K) (tick := tick) allocation m F ε := by
rw [producedTensor_eq_delete]
funext x y z
by_cases hp : passes allocation m ε 0 x ∧ passes allocation m ε 1 y ∧
passes allocation m ε 2 z
· simp only [ExactRecovery.delete, ite_eq_left hp]
unfold pulled Tensor.pullback
rw [canonical_eq_delete]
have hx := childWindows_of_produced_masks allocation m hε 0 x hp.1
have hy := childWindows_of_produced_masks allocation m hε 1 y hp.2.1
have hz := childWindows_of_produced_masks allocation m hε 2 z hp.2.2
unfold ExactRecovery.delete
rw [ite_eq_left ⟨hx, hy, hz⟩]
exact regroup_coefficient allocation m x y z
· simp only [ExactRecovery.delete, ite_eq_right hp]
def pullbackMap (F : Type*) [Field F] (ε : ℝ) :
LocalMap (AllFieldHistoryRecovery.canonical F (K := K) (tick := tick) allocation m ε)
(pulled (K := K) (tick := tick) allocation m F ε) := by
apply LocalMap.ofExists
exact ⟨_, _, _, (Tensor.pullback_eq_restrict
(regroupWords (K := K) (tick := tick) allocation m)
(regroupWords (K := K) (tick := tick) allocation m)
(regroupWords (K := K) (tick := tick) allocation m) _).symm⟩
def map (F : Type*) [Field F] {ε : ℝ} (hε : 0 ≤ ε) :
LocalMap (AllFieldHistoryRecovery.canonical F (K := K) (tick := tick) allocation m ε)
(AllFieldHistoryRegrouping.producedTensor (K := K) (tick := tick) allocation m F ε) := by
let first := pullbackMap (K := K) (tick := tick) allocation m F ε
let last := LocalMap.delete (pulled (K := K) (tick := tick) allocation m F ε)
(passes allocation m ε 0) (passes allocation m ε 1) (passes allocation m ε 2)
let combined := first.comp last
exact ⟨combined.x, combined.y, combined.z,
combined.coefficient.trans (delete_pulled allocation m F hε)⟩
end MatrixMultiplication.AllFieldHistoryProduced
end
end OAI
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