| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.TickCarry |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.SourceAssignment |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.Produced |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.SelectionCore |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.AssignmentRetention |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.Assignments |
|
|
| namespace OAI |
|
|
| /-! Finite extraction histories, inherited masks and recovery bounds. -/ |
|
|
| noncomputable section |
|
|
| namespace MatrixMultiplication.AllFieldHistoryTick |
|
|
| open MatrixMultiplication.Foundation AllFieldHistory AllFieldFiniteFamily |
| open AllFieldGroupSelection AllFieldHistoryGroupedRecovery |
| open scoped BigOperators Classical |
| attribute [local instance] Classical.propDecidable Classical.decEq |
|
|
| variable {K tick : ℕ} (allocation : Allocation) (m : ℕ) (ε slack : ℝ) |
|
|
| abbrev Choices := ∀ sigma : Placement, |
| Selected (K := K) (tick := tick) allocation m ε sigma slack |
|
|
| abbrev AmbientTag (sigma : Placement) := |
| JointCoarseHashing.Triple |
| (AllFieldGroupOrbitData.Pos (K := K) (tick := tick) allocation m sigma) |
|
|
| abbrev Tags (chosen : Choices (K := K) (tick := tick) allocation m ε slack) := |
| ∀ sigma : Placement, {e // e ∈ (chosen sigma).targets} |
|
|
| local instance tagsDecidableEq (chosen : Choices (K := K) (tick := tick) allocation m ε slack) : |
| DecidableEq (Tags allocation m ε slack chosen) := Classical.decEq _ |
|
|
| @[instance_reducible] |
| private def coarseTripleFintype (P : Type) [Fintype P] : |
| Fintype (JointCoarseHashing.Triple P) := inferInstance |
| |
| instance ambientTagFintype (sigma : Placement) : |
| Fintype (AmbientTag (K := K) (tick := tick) allocation m sigma) := |
| coarseTripleFintype (AllFieldGroupOrbitData.Pos (K := K) (tick := tick) allocation m sigma) |
| |
| instance ambientTagsFintype : |
| Fintype (∀ sigma, AmbientTag (K := K) (tick := tick) allocation m sigma) := inferInstance |
| |
| instance tagsFintype (chosen : Choices (K := K) (tick := tick) allocation m ε slack) : |
| Fintype (Tags allocation m ε slack chosen) := inferInstance |
|
|
| def select (chosen : Choices (K := K) (tick := tick) allocation m ε slack) |
| (j : Tags allocation m ε slack chosen) : |
| ∀ sigma, AmbientTag (K := K) (tick := tick) allocation m sigma := |
| fun sigma => AllFieldGroupOrbitData.coarse allocation m sigma (j sigma).val |
|
|
| theorem select_injective |
| (chosen : Choices (K := K) (tick := tick) allocation m ε slack) : |
| Function.Injective (select allocation m ε slack chosen) := by |
| intro _ _ hequal |
| funext sigma |
| apply Subtype.ext |
| exact AllFieldGroupOrbitData.coarse_injective allocation m sigma (congrFun hequal sigma) |
|
|
| def targets (chosen : Choices (K := K) (tick := tick) allocation m ε slack) |
| (j : Tags allocation m ε slack chosen) : |
| AllFieldHistoryRecovery.Targets (K := K) (tick := tick) allocation m := |
| joinGroupTargets (activeCounts allocation m) (fun sigma => (j sigma).val) |
|
|
| def assignments (chosen : Choices (K := K) (tick := tick) allocation m ε slack) |
| (sigma : Placement) := |
| AllFieldGroupAssignmentRetention.physicalAssignment allocation m ε sigma |
| (JointOrdinarySelection.hashLevel |
| (selectionHashRate (K := K) (tick := tick) allocation sigma slack) m) |
| (JointOrdinarySelection.hashSet |
| (selectionHashRate (K := K) (tick := tick) allocation sigma slack) m) |
| (chosen sigma).sample |
|
|
| def assignment (chosen : Choices (K := K) (tick := tick) allocation m ε slack) := |
| groupedAssignment allocation m (AmbientTag (K := K) (tick := tick) allocation m) |
| (assignments allocation m ε slack chosen) |
|
|
| theorem coherent (F : Type*) [Field F] |
| (chosen : Choices (K := K) (tick := tick) allocation m ε slack) : |
| JointExtraction.Coherent |
| (AllFieldHistorySource.preparedSource (K := K) (tick := tick) allocation m F ε) |
| (assignment allocation m ε slack chosen) := by |
| apply AllFieldHistorySource.groupedAssignment_coherent allocation m F ε |
| intro sigma |
| exact AllFieldGroupAssignments.physicalAssignment_coherent F allocation m ε sigma |
| _ _ (chosen sigma).sample |
|
|
| theorem repair (F : Type*) [Field F] (hε : 0 < ε) |
| (hm : AllFieldHistoryRecovery.minimumDilation (K := K) (tick := tick) allocation ε ≤ m) |
| (chosen : Choices (K := K) (tick := tick) allocation m ε slack) |
| (j : Tags allocation m ε slack chosen) : |
| InverseLinearRecovery.RecoveredBy |
| (AllFieldHistoryRecovery.ideal F allocation m ε (targets allocation m ε slack chosen j)) |
| (JointExtraction.branch |
| (AllFieldHistorySource.preparedSource (K := K) (tick := tick) allocation m F ε) |
| (assignment allocation m ε slack chosen) (select allocation m ε slack chosen j)) |
| (AllFieldHistoryRecovery.shiftCount (K := K) (tick := tick) allocation ε m) := by |
| rw [AllFieldHistorySource.preparedSource_eq_delete] |
| apply repair_grouped F allocation hε hm |
| (AmbientTag (K := K) (tick := tick) allocation m) |
| (assignments allocation m ε slack chosen) |
| (select allocation m ε slack chosen j) (fun sigma => (j sigma).val) |
| · intro sigma w hw |
| have hs := AllFieldGroupAssignmentRetention.assigned_some_idealSide |
| allocation m ε sigma _ _ (chosen sigma).sample (j sigma).val (sigma.symm 0, w) hw |
| simpa only [Equiv.apply_symm_apply] using hs |
| · intro sigma w hw |
| have hs := AllFieldGroupAssignmentRetention.assigned_some_idealSide |
| allocation m ε sigma _ _ (chosen sigma).sample (j sigma).val (sigma.symm 1, w) hw |
| simpa only [Equiv.apply_symm_apply] using hs |
| · intro sigma w hw |
| have hs := AllFieldGroupAssignmentRetention.assigned_some_idealSide |
| allocation m ε sigma _ _ (chosen sigma).sample (j sigma).val (sigma.symm 2, w) hw |
| simpa only [Equiv.apply_symm_apply] using hs |
| · intro side w hw sigma |
| have hg := (chosen sigma).good (j sigma).val (j sigma).property |
| have hh := AllFieldGroupAssignmentRetention.good_projected_failure_le F |
| allocation m ε sigma _ _ |
| (JointOrdinarySelection.hashSet_AP (selectionHashRate allocation sigma slack) m) |
| (m : ℝ) (chosen sigma).sample (targets allocation m ε slack chosen j) |
| (by simpa only [targets, project_joinGroupTargets] using hg) side w hw |
| simp only [targets, project_joinGroupTargets] at hh |
| rw [filter_eq_rejected] at hh ⊢ |
| dsimp only [assignments, select] |
| exact hh |
|
|
| def canonicalMap (F : Type*) [Field F] |
| (e : AllFieldHistoryRecovery.Targets (K := K) (tick := tick) allocation m) : |
| LocalMap (AllFieldHistoryRecovery.ideal F allocation m ε e) |
| (AllFieldHistoryRecovery.canonical (K := K) (tick := tick) F allocation m ε) := by |
| apply LocalMap.ofExists |
| refine ⟨_, _, _, (Tensor.pullback_eq_restrict |
| (AllFieldHistoryRecovery.coordinates allocation m e).symm |
| (AllFieldHistoryRecovery.coordinates allocation m e).symm |
| (AllFieldHistoryRecovery.coordinates allocation m e).symm _).symm.trans ?_⟩ |
| funext x y z |
| change AllFieldHistoryRecovery.ideal F allocation m ε e _ _ _ = _ |
| rw [AllFieldHistoryRecovery.ideal_coordinates] |
| simp only [Equiv.apply_symm_apply] |
|
|
| def activeExecution (F : Type*) [Field F] (hε : 0 < ε) |
| (hm : AllFieldHistoryRecovery.minimumDilation (K := K) (tick := tick) allocation ε ≤ m) |
| (chosen : Choices (K := K) (tick := tick) allocation m ε slack) : |
| Execution |
| (AllFieldHistoryRegrouping.activeTensor (K := K) (tick := tick) allocation m F ε) |
| (Tensor.directSum (fun _ : Tags allocation m ε slack chosen => |
| AllFieldHistoryRegrouping.producedTensor (K := K) (tick := tick) allocation m F ε)) := by |
| let prepared := (Execution.initial |
| (AllFieldHistoryRegrouping.activeTensor (K := K) (tick := tick) allocation m F ε)).restrict _ |
| (AllFieldHistorySource.preparedMap (K := K) (tick := tick) allocation m F ε) |
| let repaired := prepared.extractSelectedAndRepair |
| (assignment allocation m ε slack chosen) (coherent allocation m ε slack F chosen) |
| (select allocation m ε slack chosen) (select_injective allocation m ε slack chosen) |
| (fun j => AllFieldHistoryRecovery.ideal F allocation m ε (targets allocation m ε slack chosen j)) |
| (AllFieldHistoryRecovery.shiftCount (K := K) (tick := tick) allocation ε m) |
| (repair allocation m ε slack F hε hm chosen) |
| let canonical := repaired.restrict _ (LocalMap.directSum _ _ |
| (fun j => canonicalMap allocation m ε F (targets allocation m ε slack chosen j))) |
| exact canonical.restrict _ |
| ((AllFieldHistoryProduced.map (K := K) (tick := tick) allocation m F hε.le).parallelCopies _) |
|
|
| @[simp] theorem activeExecution_copies (F : Type*) [Field F] (hε : 0 < ε) |
| (hm : AllFieldHistoryRecovery.minimumDilation (K := K) (tick := tick) allocation ε ≤ m) |
| (chosen : Choices (K := K) (tick := tick) allocation m ε slack) : |
| Fintype.card (activeExecution allocation m ε slack F hε hm chosen).Copies = |
| 2 ^ (3 * AllFieldHistoryRecovery.shiftCount (K := K) (tick := tick) allocation ε m) := by |
| change Fintype.card (InverseLinearRecovery.MaskRectangles |
| (AllFieldHistoryRecovery.shiftCount (K := K) (tick := tick) allocation ε m) × PUnit) = _ |
| rw [Fintype.card_prod, ExactRecovery.card_mask_rectangles, Fintype.card_fin, |
| Fintype.card_punit, mul_one] |
|
|
| theorem card_tags (chosen : Choices (K := K) (tick := tick) allocation m ε slack) : |
| Fintype.card (Tags allocation m ε slack chosen) = |
| ∏ _sigma : Placement, |
| ⌊Real.exp ((m : ℝ) * (nativeCapacity (K := K) (tick := tick) allocation - slack))⌋₊ := by |
| rw [Fintype.card_pi] |
| apply Finset.prod_congr rfl |
| intro sigma _ |
| simpa only [Fintype.card_coe] using (chosen sigma).card_targets |
|
|
| def execution (F : Type*) [Field F] (hε : 0 < ε) |
| (hm : AllFieldHistoryRecovery.minimumDilation (K := K) (tick := tick) allocation ε ≤ m) |
| (chosen : Choices (K := K) (tick := tick) allocation m ε slack) : |
| Execution (stateTensor F (K := K) allocation m ε tick) |
| (Tensor.directSum (fun _ : Tags allocation m ε slack chosen => |
| stateTensor F (K := K) allocation m ε (tick + 1))) := |
| AllFieldHistoryTickCarry.liftExecution (K := K) (tick := tick) allocation m F ε |
| (activeExecution allocation m ε slack F hε hm chosen) |
|
|
| @[simp] theorem execution_copies (F : Type*) [Field F] (hε : 0 < ε) |
| (hm : AllFieldHistoryRecovery.minimumDilation (K := K) (tick := tick) allocation ε ≤ m) |
| (chosen : Choices (K := K) (tick := tick) allocation m ε slack) : |
| Fintype.card (execution allocation m ε slack F hε hm chosen).Copies = |
| 2 ^ (3 * AllFieldHistoryRecovery.shiftCount (K := K) (tick := tick) allocation ε m) := |
| activeExecution_copies allocation m ε slack F hε hm chosen |
|
|
| end MatrixMultiplication.AllFieldHistoryTick |
|
|
| end |
|
|
| end OAI |
|
|
