| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.RecoveryCore |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.GroupAssignment |
|
|
| namespace OAI |
|
|
| /-! Finite extraction histories, inherited masks and recovery bounds. -/ |
|
|
| noncomputable section |
|
|
| namespace MatrixMultiplication.AllFieldHistoryGroupMasks |
|
|
| open MatrixMultiplication.Foundation AllFieldHistory AllFieldHistorySupport AllFieldHistoryChildLaws |
| open AllFieldHistoryMasks AllFieldHistoryRawMasks AllFieldHistoryIncomingMasks |
| open JointPopulation JointCanonicalization JointCanonicalCW InheritedMasks |
| open scoped BigOperators |
| attribute [local instance] Classical.propDecidable Classical.decEq |
|
|
| variable {K tick : ℕ} |
|
|
| def groupCounts (allocation : Allocation) (m : ℕ) (sigma : Placement) |
| (h : ActiveOrder K tick sigma) : JointPopulation.Shape → ℕ := |
| activeCounts allocation m h.val |
|
|
| abbrev GroupRaw (allocation : Allocation) (m : ℕ) (sigma : Placement) := |
| RawPairs (groupCounts (K := K) (tick := tick) allocation m sigma) |
| (fun h => Fin (activeHalfLength h.val) → Fin 7) |
| (fun h => Fin (activeHalfLength h.val) → Fin 7) |
|
|
| def projectRaw (allocation : Allocation) (m : ℕ) (sigma : Placement) |
| (w : AllFieldHistoryRecovery.Raw (K := K) (tick := tick) allocation m) : |
| GroupRaw (K := K) (tick := tick) allocation m sigma := fun h j => w h.val j |
|
|
| def marginal (allocation : Allocation) (m : ℕ) (side : Fin 3) (sigma : Placement) |
| (w : GroupRaw (K := K) (tick := tick) allocation m sigma) : Prop := |
| ∀ (h : ActiveOrder K tick sigma) a, |
| wordPopulation (fun j => coarse activeHalfLength activeHalfLength |
| AllFieldHistoryRecovery.halfLength_le_eight side h.val (w h j)) a = |
| wordPopulation (sideWord (activeCounts allocation m) side |
| (triple (activeCounts allocation m) (canonicalTarget (activeCounts allocation m))) h.val) a |
|
|
| def incoming (allocation : Allocation) (m : ℕ) (sigma : Placement) |
| (h : ActiveOrder K tick sigma) (w : GroupRaw (K := K) (tick := tick) allocation m sigma) : |
| HistoryWord allocation m (h.val.val.1.source, h.val.val.2) := |
| fun p => joinHalves h.val.val.1 |
| (w h ((sourcePositionsEquiv allocation m h.val).symm p)) |
|
|
| def received (allocation : Allocation) (m : ℕ) (ε : ℝ) (side : Fin 3) |
| (sigma : Placement) (w : GroupRaw (K := K) (tick := tick) allocation m sigma) : Prop := |
| ∀ h : ActiveOrder K tick sigma, |
| residentMask allocation m ε (h.val.val.1.source, h.val.val.2) side |
| (incoming allocation m sigma h w) |
|
|
| def rawPass (allocation : Allocation) (m : ℕ) (ε : ℝ) (side : Fin 3) |
| (sigma : Placement) (w : GroupRaw (K := K) (tick := tick) allocation m sigma) : Prop := |
| ∀ (i : MaskIndex K tick) (hi : i.1.val.physicalOrder = sigma), |
| |(∑ j : JointPopulation.Positions (activeCounts allocation m) i.1, |
| if statistic i.1 (w ⟨i.1, hi⟩ j).1 = i.2.1 i.1 ∧ |
| statistic i.1 (w ⟨i.1, hi⟩ j).2 = i.2.2 i.1 then (1 : ℝ) else 0) / |
| (population allocation m (i.1.val.1.source, i.1.val.2) : ℝ) - |
| center allocation side i| ≤ pairWidth ε i.1 |
|
|
| def completeKeep (allocation : Allocation) (m : ℕ) (ε : ℝ) (side : Fin 3) |
| (sigma : Placement) (w : GroupRaw (K := K) (tick := tick) allocation m sigma) : Prop := |
| marginal allocation m side sigma w ∧ received allocation m ε side sigma w ∧ |
| rawPass allocation m ε side sigma w |
|
|
| theorem marginal_project (allocation : Allocation) (m : ℕ) (side : Fin 3) |
| (w : AllFieldHistoryRecovery.Raw (K := K) (tick := tick) allocation m) |
| (hw : JointCanonicalMixed.sideMask (activeCounts allocation m) |
| activeHalfLength activeHalfLength AllFieldHistoryRecovery.halfLength_le_eight side w) |
| (sigma : Placement) : marginal allocation m side sigma (projectRaw allocation m sigma w) := by |
| intro h a |
| exact hw h.val a |
|
|
| theorem received_project (allocation : Allocation) (m : ℕ) (ε : ℝ) (side : Fin 3) |
| (w : AllFieldHistoryRecovery.Raw (K := K) (tick := tick) allocation m) |
| (hw : AllFieldHistoryIncomingMasks.received allocation m ε side w) |
| (sigma : Placement) : received allocation m ε side sigma (projectRaw allocation m sigma w) := by |
| intro h |
| exact hw h.val |
|
|
| theorem rawPass_project (allocation : Allocation) (m : ℕ) (ε : ℝ) (side : Fin 3) |
| (w : AllFieldHistoryRecovery.Raw (K := K) (tick := tick) allocation m) |
| (hw : AllFieldHistoryRawMasks.rawPass allocation m ε side w) |
| (sigma : Placement) : rawPass allocation m ε side sigma (projectRaw allocation m sigma w) := by |
| intro i hi |
| exact hw i |
|
|
| theorem completeKeep_project (allocation : Allocation) (m : ℕ) (ε : ℝ) (side : Fin 3) |
| (w : AllFieldHistoryRecovery.Raw (K := K) (tick := tick) allocation m) |
| (hw : AllFieldHistoryRecovery.completeKeep allocation m ε side w) |
| (sigma : Placement) : completeKeep allocation m ε side sigma |
| (projectRaw allocation m sigma w) := |
| ⟨marginal_project allocation m side w hw.1 sigma, |
| received_project allocation m ε side w hw.2.1 sigma, |
| rawPass_project allocation m ε side w hw.2.2 sigma⟩ |
|
|
| theorem completeKeep_iff_all_groups (allocation : Allocation) (m : ℕ) (ε : ℝ) |
| (side : Fin 3) (w : AllFieldHistoryRecovery.Raw (K := K) (tick := tick) allocation m) : |
| AllFieldHistoryRecovery.completeKeep allocation m ε side w ↔ |
| ∀ sigma, completeKeep allocation m ε side sigma (projectRaw allocation m sigma w) := by |
| refine ⟨fun hw sigma => completeKeep_project allocation m ε side w hw sigma, ?_⟩ |
| intro hw |
| refine ⟨?_, ?_, ?_⟩ |
| · intro h a |
| exact (hw h.val.physicalOrder).1 ⟨h, rfl⟩ a |
| · intro h |
| exact (hw h.val.physicalOrder).2.1 ⟨h, rfl⟩ |
| · intro i |
| exact (hw i.1.val.physicalOrder).2.2 i rfl |
|
|
| def groupRawSource (F : Type*) [CommRing F] (allocation : Allocation) (m : ℕ) |
| (sigma : Placement) : Tensor F |
| (GroupRaw (K := K) (tick := tick) allocation m sigma) |
| (GroupRaw (K := K) (tick := tick) allocation m sigma) (GroupRaw (K := K) (tick := tick) allocation m sigma) := |
| sourceTensor (groupCounts allocation m sigma) |
| (fun h => Fin (activeHalfLength h.val) → Fin 7) |
| (fun h => Fin (activeHalfLength h.val) → Fin 7) |
| (fun h => AllFieldHistoryRecovery.parents F h.val) |
|
|
| def groupPreparedSource (F : Type*) [CommRing F] (allocation : Allocation) (m : ℕ) |
| (ε : ℝ) (sigma : Placement) : Tensor F |
| (GroupRaw (K := K) (tick := tick) allocation m sigma) |
| (GroupRaw (K := K) (tick := tick) allocation m sigma) (GroupRaw (K := K) (tick := tick) allocation m sigma) := |
| ExactRecovery.delete (groupRawSource F allocation m sigma) |
| (completeKeep allocation m ε 0 sigma) (completeKeep allocation m ε 1 sigma) |
| (completeKeep allocation m ε 2 sigma) |
|
|
| theorem rawSource_factor (F : Type*) [CommRing F] (allocation : Allocation) (m : ℕ) |
| (x y z : AllFieldHistoryRecovery.Raw (K := K) (tick := tick) allocation m) : |
| AllFieldHistoryRecovery.rawSource F allocation m x y z = |
| ∏ sigma : Placement, groupRawSource F allocation m sigma |
| (projectRaw allocation m sigma x) (projectRaw allocation m sigma y) |
| (projectRaw allocation m sigma z) := |
| prod_active_eq_prod_orders (fun h => ∏ j, |
| AllFieldHistoryRecovery.parents F h (x h j) (y h j) (z h j)) |
|
|
| theorem preparedSource_factor (F : Type*) [CommRing F] (allocation : Allocation) |
| (m : ℕ) (ε : ℝ) |
| (x y z : AllFieldHistoryRecovery.Raw (K := K) (tick := tick) allocation m) : |
| ExactRecovery.delete (AllFieldHistoryRecovery.rawSource F allocation m) |
| (AllFieldHistoryRecovery.completeKeep allocation m ε 0) |
| (AllFieldHistoryRecovery.completeKeep allocation m ε 1) |
| (AllFieldHistoryRecovery.completeKeep allocation m ε 2) x y z = |
| ∏ sigma : Placement, groupPreparedSource F allocation m ε sigma |
| (projectRaw allocation m sigma x) (projectRaw allocation m sigma y) |
| (projectRaw allocation m sigma z) := by |
| have heq : |
| (AllFieldHistoryRecovery.completeKeep allocation m ε 0 x ∧ |
| AllFieldHistoryRecovery.completeKeep allocation m ε 1 y ∧ |
| AllFieldHistoryRecovery.completeKeep allocation m ε 2 z) ↔ |
| ∀ sigma : Placement, |
| completeKeep allocation m ε 0 sigma (projectRaw allocation m sigma x) ∧ |
| completeKeep allocation m ε 1 sigma (projectRaw allocation m sigma y) ∧ |
| completeKeep allocation m ε 2 sigma (projectRaw allocation m sigma z) := by |
| rw [completeKeep_iff_all_groups, completeKeep_iff_all_groups, completeKeep_iff_all_groups] |
| exact ⟨fun h sigma => ⟨h.1 sigma, h.2.1 sigma, h.2.2 sigma⟩, |
| fun h => ⟨fun sigma => (h sigma).1, fun sigma => (h sigma).2.1, |
| fun sigma => (h sigma).2.2⟩⟩ |
| by_cases h : AllFieldHistoryRecovery.completeKeep allocation m ε 0 x ∧ |
| AllFieldHistoryRecovery.completeKeep allocation m ε 1 y ∧ |
| AllFieldHistoryRecovery.completeKeep allocation m ε 2 z |
| · rw [ExactRecovery.delete, ite_eq_left h, rawSource_factor] |
| apply Finset.prod_congr rfl |
| intro sigma _ |
| exact (ite_eq_left (heq.mp h sigma)).symm |
| · rw [ExactRecovery.delete, ite_eq_right h] |
| have hn : ¬∀ sigma : Placement, |
| completeKeep allocation m ε 0 sigma (projectRaw allocation m sigma x) ∧ |
| completeKeep allocation m ε 1 sigma (projectRaw allocation m sigma y) ∧ |
| completeKeep allocation m ε 2 sigma (projectRaw allocation m sigma z) := |
| fun hh => h (heq.mpr hh) |
| obtain ⟨sigma, hs⟩ := not_forall.mp hn |
| exact (Finset.prod_eq_zero (Finset.mem_univ sigma) (ite_eq_right hs)).symm |
|
|
| end MatrixMultiplication.AllFieldHistoryGroupMasks |
|
|
| end |
|
|
| end OAI |
|
|
