| 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 |
|
|
