| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.Regrouping |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.IncomingMasks |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.RecoveryCore |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.GroupMasks |
|
|
| namespace OAI |
|
|
| /-! Finite extraction histories, inherited masks and recovery bounds. -/ |
|
|
| noncomputable section |
|
|
| namespace MatrixMultiplication.AllFieldHistorySource |
|
|
| open MatrixMultiplication.Foundation AllFieldHistory AllFieldFiniteFamily |
| open AllFieldHistoryRegrouping AllFieldHistoryIncomingMasks |
| open scoped BigOperators Classical |
| attribute [local instance] Classical.propDecidable Classical.decEq |
|
|
| private def deleteWithPred {F : Type*} [Field F] {X Y Z : Type} |
| [Fintype X] [Fintype Y] [Fintype Z] |
| (source : Tensor F X Y Z) (px : X → Prop) (py : Y → Prop) (pz : Z → Prop) |
| [DecidablePred px] [DecidablePred py] [DecidablePred pz] : |
| LocalMap source (ExactRecovery.delete source px py pz) := by |
| let d := LocalMap.delete source px py pz |
| refine ⟨d.x, d.y, d.z, d.coefficient.trans ?_⟩ |
| funext x y z |
| unfold ExactRecovery.delete |
| split_ifs <;> rfl |
| |
| variable {K tick : ℕ} (allocation : Allocation) (m : ℕ) |
|
|
| abbrev Raw := AllFieldHistoryRecovery.Raw (K := K) (tick := tick) allocation m |
|
|
| def toActive (w : Raw (K := K) (tick := tick) allocation m) : |
| ActiveWords (K := K) (tick := tick) allocation m := fun h => incomingWord allocation m h w |
|
|
| theorem parent_coefficient (F : Type*) [Field F] (h : Active K tick) |
| (x y z : Raw (K := K) (tick := tick) allocation m) : |
| Tensor.power |
| (CWStrands.shapeTensor (Fin (currentLength h.val.1.source)) |
| (currentPhysicalShape (h.val.1.source, h.val.2))) |
| (population allocation m (h.val.1.source, h.val.2)) |
| (incomingWord allocation m h x) (incomingWord allocation m h y) |
| (incomingWord allocation m h z) = |
| ∏ i : JointPopulation.Positions (activeCounts allocation m) h, |
| AllFieldHistoryRecovery.parents F h (x h i) (y h i) (z h i) := by |
| change (∏ p : Fin (population allocation m (h.val.1.source, h.val.2)), |
| CWStrands.shapeTensor _ _ (incomingWord allocation m h x p) |
| (incomingWord allocation m h y p) (incomingWord allocation m h z p)) = _ |
| refine ((sourcePositionsEquiv allocation m h).prod_comp (fun p => |
| CWStrands.shapeTensor (Fin (currentLength h.val.1.source)) |
| (currentPhysicalShape (h.val.1.source, h.val.2)) |
| (incomingWord allocation m h x p) (incomingWord allocation m h y p) |
| (incomingWord allocation m h z p))).symm.trans ?_ |
| apply Finset.prod_congr rfl |
| intro i _ |
| simp only [incomingWord, Equiv.symm_apply_apply] |
| erw [shapeTensor_joinHalves] |
| unfold AllFieldHistoryRecovery.parents JointCanonicalMixed.inputTensor |
| rw [ite_eq_left rfl] |
| unfold JointCanonicalCW.parentTensor currentPhysicalShape |
| rw [source_shape] |
| rfl |
|
|
| def receivedSource (F : Type*) [Field F] (ε : ℝ) : |
| Tensor F (Raw (K := K) (tick := tick) allocation m) (Raw (K := K) (tick := tick) allocation m) (Raw (K := K) (tick := tick) allocation m) := |
| ExactRecovery.delete (AllFieldHistoryRecovery.rawSource F (K := K) (tick := tick) allocation m) |
| (received allocation m ε 0) (received allocation m ε 1) (received allocation m ε 2) |
|
|
| theorem received_coefficient (F : Type*) [Field F] (ε : ℝ) |
| (x y z : Raw (K := K) (tick := tick) allocation m) : |
| activeTensor (K := K) (tick := tick) allocation m F ε (toActive allocation m x) (toActive allocation m y) |
| (toActive allocation m z) = receivedSource (K := K) (tick := tick) allocation m F ε x y z := by |
| change (∏ h : Active K tick, historyTensor F allocation m ε |
| (h.val.1.source, h.val.2) (incomingWord allocation m h x) |
| (incomingWord allocation m h y) (incomingWord allocation m h z)) = _ |
| unfold receivedSource ExactRecovery.delete |
| by_cases hp : received allocation m ε 0 x ∧ received allocation m ε 1 y ∧ |
| received allocation m ε 2 z |
| · rw [ite_eq_left hp] |
| change _ = ∏ h : Active K tick, ∏ i : JointPopulation.Positions (activeCounts allocation m) h, |
| AllFieldHistoryRecovery.parents F h (x h i) (y h i) (z h i) |
| apply Finset.prod_congr rfl |
| intro h _ |
| unfold historyTensor ExactRecovery.delete |
| rw [ite_eq_left ⟨hp.1 h, hp.2.1 h, hp.2.2 h⟩] |
| exact parent_coefficient allocation m F h x y z |
| · rw [ite_eq_right hp] |
| have hh : ∃ h : Active K tick, |
| ¬(residentMask allocation m ε (h.val.1.source, h.val.2) 0 (incomingWord allocation m h x) ∧ |
| residentMask allocation m ε (h.val.1.source, h.val.2) 1 (incomingWord allocation m h y) ∧ |
| residentMask allocation m ε (h.val.1.source, h.val.2) 2 (incomingWord allocation m h z)) := by |
| by_contra hn |
| apply hp |
| push Not at hn |
| exact ⟨fun h => (hn h).1, fun h => (hn h).2.1, fun h => (hn h).2.2⟩ |
| obtain ⟨h, hh⟩ := hh |
| apply Finset.prod_eq_zero (Finset.mem_univ h) |
| exact ite_eq_right hh |
|
|
| def receivedMap (F : Type*) [Field F] (ε : ℝ) : |
| LocalMap (activeTensor (K := K) (tick := tick) allocation m F ε) |
| (receivedSource (K := K) (tick := tick) allocation m F ε) := by |
| apply LocalMap.ofExists |
| refine ⟨_, _, _, (Tensor.pullback_eq_restrict |
| (toActive (K := K) (tick := tick) allocation m) |
| (toActive (K := K) (tick := tick) allocation m) |
| (toActive (K := K) (tick := tick) allocation m) _).symm.trans ?_⟩ |
| funext x y z |
| exact received_coefficient allocation m F ε x y z |
|
|
| def marginal (side : Fin 3) (w : Raw (K := K) (tick := tick) allocation m) : Prop := |
| JointCanonicalMixed.sideMask (activeCounts allocation m) activeHalfLength activeHalfLength |
| AllFieldHistoryRecovery.halfLength_le_eight side w |
|
|
| def preparedSource (F : Type*) [Field F] (ε : ℝ) : |
| Tensor F (Raw (K := K) (tick := tick) allocation m) (Raw (K := K) (tick := tick) allocation m) (Raw (K := K) (tick := tick) allocation m) := |
| ExactRecovery.delete (receivedSource (K := K) (tick := tick) allocation m F ε) |
| (fun x => marginal (K := K) (tick := tick) allocation m 0 x ∧ AllFieldHistoryRawMasks.rawPass allocation m ε 0 x) |
| (fun y => marginal (K := K) (tick := tick) allocation m 1 y ∧ AllFieldHistoryRawMasks.rawPass allocation m ε 1 y) |
| (fun z => marginal (K := K) (tick := tick) allocation m 2 z ∧ AllFieldHistoryRawMasks.rawPass allocation m ε 2 z) |
|
|
| theorem preparedSource_eq_delete (F : Type*) [Field F] (ε : ℝ) : |
| preparedSource (K := K) (tick := tick) allocation m F ε = |
| ExactRecovery.delete (AllFieldHistoryRecovery.rawSource F (K := K) (tick := tick) allocation m) |
| (AllFieldHistoryRecovery.completeKeep allocation m ε 0) |
| (AllFieldHistoryRecovery.completeKeep allocation m ε 1) |
| (AllFieldHistoryRecovery.completeKeep allocation m ε 2) := by |
| funext x y z |
| unfold preparedSource receivedSource marginal AllFieldHistoryRecovery.completeKeep |
| ExactRecovery.delete |
| split_ifs <;> simp_all only [and_true, true_and, not_false_eq_true] |
| all_goals tauto |
| |
| def preparedMap (F : Type*) [Field F] (ε : ℝ) : |
| LocalMap (activeTensor (K := K) (tick := tick) allocation m F ε) |
| (preparedSource (K := K) (tick := tick) allocation m F ε) := |
| (receivedMap (K := K) (tick := tick) allocation m F ε).comp (deleteWithPred (receivedSource (K := K) (tick := tick) allocation m F ε) |
| (fun x => marginal (K := K) (tick := tick) allocation m 0 x ∧ AllFieldHistoryRawMasks.rawPass allocation m ε 0 x) |
| (fun y => marginal (K := K) (tick := tick) allocation m 1 y ∧ AllFieldHistoryRawMasks.rawPass allocation m ε 1 y) |
| (fun z => marginal (K := K) (tick := tick) allocation m 2 z ∧ AllFieldHistoryRawMasks.rawPass allocation m ε 2 z)) |
|
|
| abbrev GroupWords := ∀ sigma : Placement, |
| AllFieldHistoryGroupMasks.GroupRaw (K := K) (tick := tick) allocation m sigma |
|
|
| def groupsEquiv : Raw (K := K) (tick := tick) allocation m ≃ |
| GroupWords (K := K) (tick := tick) allocation m := |
| activeWordsEquivOrders (fun h : Active K tick => |
| JointPopulation.Positions (activeCounts allocation m) h → |
| ((Fin (activeHalfLength h) → Fin 7) × (Fin (activeHalfLength h) → Fin 7))) |
|
|
| @[simp] theorem project_groups_symm |
| (x : GroupWords (K := K) (tick := tick) allocation m) (sigma : Placement) : |
| AllFieldHistoryGroupMasks.projectRaw allocation m sigma |
| ((groupsEquiv allocation m).symm x) = x sigma := |
| congrFun ((groupsEquiv allocation m).apply_symm_apply x) sigma |
|
|
| theorem grouped_coefficient (F : Type*) [Field F] (ε : ℝ) |
| (x y z : GroupWords (K := K) (tick := tick) allocation m) : |
| preparedSource (K := K) (tick := tick) allocation m F ε |
| ((groupsEquiv allocation m).symm x) ((groupsEquiv allocation m).symm y) |
| ((groupsEquiv allocation m).symm z) = |
| CommonDimensions.familyProduct |
| (fun sigma => AllFieldHistoryGroupMasks.groupPreparedSource F allocation m ε sigma) |
| x y z := by |
| rw [preparedSource_eq_delete, AllFieldHistoryGroupMasks.preparedSource_factor] |
| simp only [project_groups_symm, CommonDimensions.familyProduct] |
|
|
| def groupsMap (F : Type*) [Field F] (ε : ℝ) : |
| LocalMap (preparedSource (K := K) (tick := tick) allocation m F ε) |
| (CommonDimensions.familyProduct (fun sigma : Placement => |
| AllFieldHistoryGroupMasks.groupPreparedSource F (K := K) (tick := tick) |
| allocation m ε sigma)) := by |
| apply LocalMap.ofExists |
| refine ⟨_, _, _, (Tensor.pullback_eq_restrict |
| (groupsEquiv (K := K) (tick := tick) allocation m).symm |
| (groupsEquiv (K := K) (tick := tick) allocation m).symm |
| (groupsEquiv (K := K) (tick := tick) allocation m).symm _).symm.trans ?_⟩ |
| funext x y z |
| exact grouped_coefficient allocation m F ε x y z |
|
|
| end MatrixMultiplication.AllFieldHistorySource |
|
|
| end |
|
|
| end OAI |
|
|
