| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.RecoveryCore |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.Masks |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.RecoveryBudget |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.RawMasks |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.IncomingMasks |
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.CanonicalSymmetry |
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.OrbitRecovery |
|
|
| namespace OAI |
|
|
| /-! Finite extraction histories, inherited masks and recovery bounds. -/ |
|
|
| noncomputable section |
|
|
| namespace MatrixMultiplication.AllFieldHistoryRecovery |
|
|
| open MatrixMultiplication.Foundation AllFieldHistory AllFieldHistoryChildLaws |
| open AllFieldHistorySupport JointPopulation JointCanonicalization JointCanonicalCW |
| open PermutationMatching |
| open scoped BigOperators |
|
|
| attribute [local instance] Classical.propDecidable Classical.decEq |
|
|
| variable {K tick : ℕ} |
|
|
| theorem ideal_coordinates (F : Type*) [CommRing F] (allocation : Allocation) |
| (m : ℕ) (ε : ℝ) (e : Targets (K := K) (tick := tick) allocation m) |
| (x y z : Raw (K := K) (tick := tick) allocation m) : |
| ideal F allocation m ε e x y z = canonical F allocation m ε |
| (coordinates allocation m e x) (coordinates allocation m e y) |
| (coordinates allocation m e z) := |
| ideal_eq_canonicalIdeal (activeCounts allocation m) (Left activeHalfLength) |
| (Right activeHalfLength) (parents F) |
| (coarse activeHalfLength activeHalfLength halfLength_le_eight) |
| childStatistic childStatistic leftPrescription rightPrescription |
| (outputWidth ε) (outputWidth ε) e x y z |
|
|
| theorem ideal_support_windows (F : Type*) [CommRing F] (allocation : Allocation) |
| (m : ℕ) (ε : ℝ) (e : Targets (K := K) (tick := tick) allocation m) |
| (x y z : Raw (K := K) (tick := tick) allocation m) (hq : ideal F allocation m ε e x y z ≠ 0) : |
| AllFieldHistoryMasks.childWindows allocation m ε 0 (coordinates allocation m e x) ∧ |
| AllFieldHistoryMasks.childWindows allocation m ε 1 (coordinates allocation m e y) ∧ |
| AllFieldHistoryMasks.childWindows allocation m ε 2 (coordinates allocation m e z) := by |
| by_contra hn |
| apply hq |
| rw [ideal_coordinates] |
| change (if AllFieldHistoryMasks.childWindows allocation m ε 0 (coordinates allocation m e x) ∧ |
| AllFieldHistoryMasks.childWindows allocation m ε 1 (coordinates allocation m e y) ∧ |
| AllFieldHistoryMasks.childWindows allocation m ε 2 (coordinates allocation m e z) |
| then _ else 0) = 0 |
| exact ite_eq_right hn |
|
|
| theorem raw_pair_loss_le (allocation : Allocation) {m : ℕ} (hm : 2 ≤ m) |
| {ε : ℝ} (hε : 0 < ε) (side : Fin 3) |
| (e : Targets (K := K) (tick := tick) allocation m) (w : Raw (K := K) (tick := tick) allocation m) |
| (hw : AllFieldHistoryMasks.childWindows allocation m ε side |
| (coordinates allocation m e w)) : |
| letI : MulAction (Symmetries (K := K) (tick := tick) allocation m) (Raw (K := K) (tick := tick) allocation m) := rawAction allocation m e |
| (((OrbitCounting.orbitSet (G := Symmetries (K := K) (tick := tick) allocation m) w).filter |
| (fun v => ¬AllFieldHistoryRawMasks.rawPass allocation m ε side v)).card : ℝ) / |
| (OrbitCounting.orbitSet (G := Symmetries (K := K) (tick := tick) allocation m) w).card ≤ |
| AllFieldHistoryMasks.lossConstant (K := K) (tick := tick) allocation ε / m := by |
| let _ : MulAction (Symmetries (K := K) (tick := tick) allocation m) (Raw (K := K) (tick := tick) allocation m) := rawAction allocation m e |
| have htransport := EquivOrbitTransport.rejected_fraction |
| (G := Symmetries (K := K) (tick := tick) allocation m) (coordinates allocation m e) |
| (fun v => ¬AllFieldHistoryRawMasks.rawPass allocation m ε side v) w |
| have hcanonical := orbit_bad_fraction_eq_fullOrbitFilter_of_iff |
| (G := Symmetries (K := K) (tick := tick) allocation m) |
| (coordinates allocation m e w) |
| ((fun v => ¬AllFieldHistoryRawMasks.rawPass allocation m ε side v) ∘ |
| (coordinates allocation m e).symm) |
| (fun v => ∃ i : AllFieldHistoryMasks.MaskIndex K tick, |
| selectedWordPairMaskBad (AllFieldHistoryMasks.testedClasses allocation i) |
| AllFieldHistoryMasks.classStatistic AllFieldHistoryMasks.classStatistic |
| (AllFieldHistoryMasks.leftSymbol i) (AllFieldHistoryMasks.rightSymbol i) |
| (AllFieldHistoryMasks.pairWidth ε i.1) (AllFieldHistoryMasks.center allocation side i) v) |
| (fun v => by |
| change ¬AllFieldHistoryRawMasks.rawPass allocation m ε side |
| ((coordinates allocation m e).symm v) ↔ _ |
| rw [← AllFieldHistoryRawMasks.passes_pairEquiv_iff_rawPass allocation m ε side e] |
| change ¬AllFieldHistoryMasks.passes allocation m ε side |
| ((coordinates allocation m e) ((coordinates allocation m e).symm v)) ↔ _ |
| rw [Equiv.apply_symm_apply] |
| simp only [AllFieldHistoryMasks.passes, not_forall, not_not]) |
| exact htransport.trans_le (hcanonical.trans_le |
| (AllFieldHistoryMasks.full_filter_loss_le allocation hm hε side |
| (coordinates allocation m e w) hw)) |
|
|
| theorem ideal_support_marginals (F : Type*) [CommRing F] (allocation : Allocation) |
| (m : ℕ) (ε : ℝ) (e : Targets (K := K) (tick := tick) allocation m) |
| (x y z : Raw (K := K) (tick := tick) allocation m) (hq : ideal F allocation m ε e x y z ≠ 0) : |
| JointCanonicalMixed.sideMask (activeCounts allocation m) |
| activeHalfLength activeHalfLength halfLength_le_eight 0 x ∧ |
| JointCanonicalMixed.sideMask (activeCounts allocation m) |
| activeHalfLength activeHalfLength halfLength_le_eight 1 y ∧ |
| JointCanonicalMixed.sideMask (activeCounts allocation m) |
| activeHalfLength activeHalfLength halfLength_le_eight 2 z := by |
| exact JointCanonicalMixed.ideal_support_masks (activeCounts allocation m) |
| activeHalfLength activeHalfLength activeParentShape (fun _ => true) |
| halfLength_le_eight e _ _ _ x y z hq |
|
|
| def sideWeights (allocation : Allocation) (m : ℕ) (side : Fin 3) |
| (e : Targets (K := K) (tick := tick) allocation m) (w : Raw (K := K) (tick := tick) allocation m) : Prop := |
| ∀ h j, |
| CWStrands.weight (w h j).1 = (shapeSide side ((e h).val j)).val ∧ |
| CWStrands.weight (w h j).2 = activeParentShape h side - |
| (shapeSide side ((e h).val j)).val |
|
|
| theorem ideal_support_weights (F : Type*) [CommRing F] (allocation : Allocation) |
| (m : ℕ) (ε : ℝ) (e : Targets (K := K) (tick := tick) allocation m) |
| (x y z : Raw (K := K) (tick := tick) allocation m) (hq : ideal F allocation m ε e x y z ≠ 0) : |
| sideWeights allocation m 0 e x ∧ sideWeights allocation m 1 e y ∧ |
| sideWeights allocation m 2 e z := by |
| have hb : coarseIdeal (activeCounts allocation m) |
| (Left activeHalfLength) (Right activeHalfLength) (parents F) |
| (coarse activeHalfLength activeHalfLength halfLength_le_eight) e x y z ≠ 0 := by |
| intro hz |
| apply hq |
| simp only [ideal, JointCanonicalization.ideal, ExactRecovery.delete, hz, ite_self] |
| have hc : |
| coarseMask (activeCounts allocation m) (Left activeHalfLength) (Right activeHalfLength) |
| (coarse activeHalfLength activeHalfLength halfLength_le_eight) 0 e x ∧ |
| coarseMask (activeCounts allocation m) (Left activeHalfLength) (Right activeHalfLength) |
| (coarse activeHalfLength activeHalfLength halfLength_le_eight) 1 e y ∧ |
| coarseMask (activeCounts allocation m) (Left activeHalfLength) (Right activeHalfLength) |
| (coarse activeHalfLength activeHalfLength halfLength_le_eight) 2 e z := by |
| by_contra hn |
| exact hb (by simp only [coarseIdeal, ExactRecovery.delete, ite_eq_right hn]) |
| have hs : rawSource F allocation m x y z ≠ 0 := by |
| intro hz |
| apply hb |
| change (if _ then rawSource F allocation m x y z else 0) = 0 |
| simp only [hz, ite_self] |
| have ht (h : Active K tick) (j : JointPopulation.Positions (activeCounts allocation m) h) : |
| parents F h (x h j) (y h j) (z h j) ≠ 0 := by |
| intro hz |
| apply hs |
| unfold rawSource sourceTensor InheritedMasks.classProduct |
| exact Finset.prod_eq_zero (Finset.mem_univ h) |
| (Finset.prod_eq_zero (Finset.mem_univ j) hz) |
| have hp (h : Active K tick) (j : JointPopulation.Positions (activeCounts allocation m) h) : |
| CWStrands.weight (x h j).1 + CWStrands.weight (x h j).2 = activeParentShape h 0 ∧ |
| CWStrands.weight (y h j).1 + CWStrands.weight (y h j).2 = activeParentShape h 1 ∧ |
| CWStrands.weight (z h j).1 + CWStrands.weight (z h j).2 = activeParentShape h 2 := by |
| have hh := ht h j |
| change CWStrands.shapeTensor |
| (Fin (activeHalfLength h) ⊕ Fin (activeHalfLength h)) (activeParentShape h) |
| (Sum.elim (x h j).1 (x h j).2) (Sum.elim (y h j).1 (y h j).2) |
| (Sum.elim (z h j).1 (z h j).2) ≠ 0 at hh |
| simpa only [CWStrands.weight_sum_elim] using |
| CWStrands.shapeTensor_support (activeParentShape h) _ _ _ hh |
| have hx (h : Active K tick) (j : JointPopulation.Positions (activeCounts allocation m) h) : |
| CWStrands.weight (x h j).1 = (shapeSide 0 ((e h).val j)).val := |
| congrArg Fin.val (hc.1 h j) |
| have hy (h : Active K tick) (j : JointPopulation.Positions (activeCounts allocation m) h) : |
| CWStrands.weight (y h j).1 = (shapeSide 1 ((e h).val j)).val := |
| congrArg Fin.val (hc.2.1 h j) |
| have hz (h : Active K tick) (j : JointPopulation.Positions (activeCounts allocation m) h) : |
| CWStrands.weight (z h j).1 = (shapeSide 2 ((e h).val j)).val := |
| congrArg Fin.val (hc.2.2 h j) |
| refine ⟨?_, ?_, ?_⟩ |
| · intro h j |
| refine ⟨hx h j, ?_⟩ |
| have hh := (hp h j).1 |
| rw [hx h j] at hh |
| omega |
| · intro h j |
| refine ⟨hy h j, ?_⟩ |
| have hh := (hp h j).2.1 |
| rw [hy h j] at hh |
| omega |
| · intro h j |
| refine ⟨hz h j, ?_⟩ |
| have hh := (hp h j).2.2 |
| rw [hz h j] at hh |
| omega |
|
|
| theorem stageCWeights_of_sideWeights (allocation : Allocation) (m : ℕ) (side : Fin 3) |
| (e : Targets (K := K) (tick := tick) allocation m) (w : Raw (K := K) (tick := tick) allocation m) |
| (hw : sideWeights allocation m side e w) : |
| AllFieldHistoryIncomingMasks.stageCWeights allocation m side e w := by |
| intro h phi ht j |
| have hh := hw (AllFieldHistoryStageCIncoming.stageCActive h phi ht) j |
| have hleft := Fin.sum_univ_one (fun i : Fin 1 => CWLeafStatistics.weight |
| ((w (AllFieldHistoryStageCIncoming.stageCActive h phi ht) j).1 i)) |
| have hright := Fin.sum_univ_one (fun i : Fin 1 => CWLeafStatistics.weight |
| ((w (AllFieldHistoryStageCIncoming.stageCActive h phi ht) j).2 i)) |
| exact ⟨hleft.symm.trans hh.1, hright.symm.trans hh.2⟩ |
|
|
| theorem completeKeep_of_supported (allocation : Allocation) {m : ℕ} (hm : 0 < m) |
| {ε : ℝ} (hε : 0 ≤ ε) (side : Fin 3) |
| (e : Targets (K := K) (tick := tick) allocation m) (w : Raw (K := K) (tick := tick) allocation m) |
| (hmarginal : JointCanonicalMixed.sideMask (activeCounts allocation m) |
| activeHalfLength activeHalfLength halfLength_le_eight side w) |
| (hweights : sideWeights allocation m side e w) |
| (hpass : AllFieldHistoryRawMasks.rawPass allocation m ε side w) : |
| completeKeep allocation m ε side w := |
| ⟨hmarginal, AllFieldHistoryIncomingMasks.received_of_rawPass allocation hm hε side e w |
| hpass (stageCWeights_of_sideWeights allocation m side e w hweights), hpass⟩ |
|
|
| theorem occupied_split_le (allocation : Allocation) (m : ℕ) |
| (h : Active K tick) (u : JointPopulation.Shape) |
| (hu : 0 < activeCounts allocation m h u) : |
| ∀ s, shapeNat u s ≤ activeParentShape h s := |
| jointCounts_support_le allocation m h.val u hu |
|
|
| theorem ideal_smul (F : Type*) [CommRing F] (allocation : Allocation) (m : ℕ) (ε : ℝ) |
| (e : Targets (K := K) (tick := tick) allocation m) : |
| letI : MulAction (Symmetries (K := K) (tick := tick) allocation m) (Raw (K := K) (tick := tick) allocation m) := rawAction allocation m e |
| ∀ (g : Symmetries (K := K) (tick := tick) allocation m) x y z, |
| ideal F allocation m ε e (g • x) (g • y) (g • z) = ideal F allocation m ε e x y z := by |
| exact JointCanonicalSymmetry.rawIdeal_smul (activeCounts allocation m) |
| activeHalfLength activeHalfLength activeParentShape (fun _ => true) |
| halfLength_le_eight childStatistic childStatistic leftPrescription rightPrescription |
| (outputWidth ε) (outputWidth ε) |
| (fun h u hu _ => occupied_split_le allocation m h u hu) e |
|
|
| def Used (F : Type*) [CommRing F] (allocation : Allocation) (m : ℕ) (ε : ℝ) |
| (e : Targets (K := K) (tick := tick) allocation m) (side : Fin 3) |
| (w : Raw (K := K) (tick := tick) allocation m) : Prop := |
| ∃ x y z, ideal F allocation m ε e x y z ≠ 0 ∧ sideVariable side x y z = w |
|
|
| theorem used_windows (F : Type*) [CommRing F] (allocation : Allocation) |
| (m : ℕ) (ε : ℝ) (e : Targets (K := K) (tick := tick) allocation m) |
| (side : Fin 3) (w : Raw (K := K) (tick := tick) allocation m) (hw : Used F allocation m ε e side w) : |
| AllFieldHistoryMasks.childWindows allocation m ε side (coordinates allocation m e w) := by |
| obtain ⟨x, y, z, hq, rfl⟩ := hw |
| have hh := ideal_support_windows F allocation m ε e x y z hq |
| fin_cases side |
| · exact hh.1 |
| · exact hh.2.1 |
| · exact hh.2.2 |
|
|
| theorem used_keep_iff_rawPass (F : Type*) [CommRing F] (allocation : Allocation) |
| {m : ℕ} (hm : 0 < m) {ε : ℝ} (hε : 0 ≤ ε) |
| (e : Targets (K := K) (tick := tick) allocation m) |
| (side : Fin 3) (w : Raw (K := K) (tick := tick) allocation m) (hw : Used F allocation m ε e side w) : |
| completeKeep allocation m ε side w ↔ AllFieldHistoryRawMasks.rawPass allocation m ε side w := by |
| refine ⟨fun h => h.2.2, ?_⟩ |
| intro hp |
| obtain ⟨x, y, z, hq, rfl⟩ := hw |
| have hmg := ideal_support_marginals F allocation m ε e x y z hq |
| have hwt := ideal_support_weights F allocation m ε e x y z hq |
| fin_cases side |
| · exact completeKeep_of_supported allocation hm hε 0 e x hmg.1 hwt.1 hp |
| · exact completeKeep_of_supported allocation hm hε 1 e y hmg.2.1 hwt.2.1 hp |
| · exact completeKeep_of_supported allocation hm hε 2 e z hmg.2.2 hwt.2.2 hp |
|
|
| theorem used_smul (F : Type*) [CommRing F] (allocation : Allocation) |
| (m : ℕ) (ε : ℝ) (e : Targets (K := K) (tick := tick) allocation m) |
| (side : Fin 3) (w : Raw (K := K) (tick := tick) allocation m) : |
| letI : MulAction (Symmetries (K := K) (tick := tick) allocation m) (Raw (K := K) (tick := tick) allocation m) := rawAction allocation m e |
| ∀ g : Symmetries (K := K) (tick := tick) allocation m, Used F allocation m ε e side w → |
| Used F allocation m ε e side (g • w) := by |
| let _ : MulAction (Symmetries (K := K) (tick := tick) allocation m) (Raw (K := K) (tick := tick) allocation m) := rawAction allocation m e |
| intro g hw |
| obtain ⟨x, y, z, hq, rfl⟩ := hw |
| refine ⟨g • x, g • y, g • z, ?_, ?_⟩ |
| · rwa [ideal_smul] |
| · fin_cases side <;> rfl |
|
|
| theorem deterministic_loss_le (F : Type*) [CommRing F] (allocation : Allocation) |
| {m : ℕ} (hm : 2 ≤ m) {ε : ℝ} (hε : 0 < ε) |
| (e : Targets (K := K) (tick := tick) allocation m) |
| (side : Fin 3) (w : Raw (K := K) (tick := tick) allocation m) (hw : Used F allocation m ε e side w) : |
| letI : MulAction (Symmetries (K := K) (tick := tick) allocation m) (Raw (K := K) (tick := tick) allocation m) := rawAction allocation m e |
| (((OrbitCounting.orbitSet (G := Symmetries (K := K) (tick := tick) allocation m) w).filter |
| (fun v => ¬completeKeep allocation m ε side v)).card : ℝ) / |
| (OrbitCounting.orbitSet (G := Symmetries (K := K) (tick := tick) allocation m) w).card ≤ |
| AllFieldHistoryMasks.lossConstant (K := K) (tick := tick) allocation ε / m := by |
| let _ : MulAction (Symmetries (K := K) (tick := tick) allocation m) (Raw (K := K) (tick := tick) allocation m) := rawAction allocation m e |
| have hmpos : 0 < m := lt_of_lt_of_le (by decide : 0 < 2) hm |
| have heq : ((OrbitCounting.orbitSet (G := Symmetries (K := K) (tick := tick) allocation m) w).filter |
| (fun v => ¬completeKeep allocation m ε side v)) = |
| ((OrbitCounting.orbitSet (G := Symmetries (K := K) (tick := tick) allocation m) w).filter |
| (fun v => ¬AllFieldHistoryRawMasks.rawPass allocation m ε side v)) := by |
| apply Finset.filter_congr |
| intro v hv |
| obtain ⟨g, _, rfl⟩ := Finset.mem_image.mp hv |
| rw [used_keep_iff_rawPass F allocation hmpos hε.le e side (g • w) (used_smul F allocation m ε e side w g hw)] |
| rw [heq] |
| exact raw_pair_loss_le allocation hm hε side e w |
| (used_windows F allocation m ε e side w hw) |
|
|
| def holedIdeal (F : Type*) [CommRing F] (allocation : Allocation) (m : ℕ) (ε : ℝ) |
| (e : Targets (K := K) (tick := tick) allocation m) |
| (assigned : Fin 3 → Raw (K := K) (tick := tick) allocation m → Prop) : |
| 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 (ideal F allocation m ε e) |
| (fun w => assigned 0 w ∧ completeKeep allocation m ε 0 w) |
| (fun w => assigned 1 w ∧ completeKeep allocation m ε 1 w) |
| (fun w => assigned 2 w ∧ completeKeep allocation m ε 2 w) |
|
|
| theorem ideal_eq_delete_source (F : Type*) [CommRing F] (allocation : Allocation) |
| (m : ℕ) (ε : ℝ) (e : Targets (K := K) (tick := tick) allocation m) : |
| ideal F allocation m ε e = ExactRecovery.delete (rawSource F allocation m) |
| (idealSide allocation m ε 0 e) (idealSide allocation m ε 1 e) |
| (idealSide allocation m ε 2 e) := by |
| have hdelete (cx cy cz wx wy wz : Prop) (q : F) |
| [dw : Decidable (wx ∧ wy ∧ wz)] [dc : Decidable (cx ∧ cy ∧ cz)] |
| (dk : Decidable ((cx ∧ wx) ∧ (cy ∧ wy) ∧ (cz ∧ wz))) : |
| @ite F (wx ∧ wy ∧ wz) dw (@ite F (cx ∧ cy ∧ cz) dc q 0) 0 = |
| @ite F ((cx ∧ wx) ∧ (cy ∧ wy) ∧ (cz ∧ wz)) dk q 0 := by |
| let _ : Decidable ((cx ∧ wx) ∧ (cy ∧ wy) ∧ (cz ∧ wz)) := dk |
| by_cases hw : wx ∧ wy ∧ wz |
| · by_cases hc : cx ∧ cy ∧ cz |
| · have hk : (cx ∧ wx) ∧ (cy ∧ wy) ∧ (cz ∧ wz) := |
| ⟨⟨hc.1, hw.1⟩, ⟨hc.2.1, hw.2.1⟩, ⟨hc.2.2, hw.2.2⟩⟩ |
| rw [ite_eq_left hw, ite_eq_left hc, ite_eq_left hk] |
| · have hk : ¬((cx ∧ wx) ∧ (cy ∧ wy) ∧ (cz ∧ wz)) := |
| fun h => hc ⟨h.1.1, h.2.1.1, h.2.2.1⟩ |
| rw [ite_eq_left hw, ite_eq_right hc, ite_eq_right hk] |
| · have hk : ¬((cx ∧ wx) ∧ (cy ∧ wy) ∧ (cz ∧ wz)) := |
| fun h => hw ⟨h.1.2, h.2.1.2, h.2.2.2⟩ |
| rw [ite_eq_right hw, ite_eq_right hk] |
| funext x y z |
| simp only [ideal, JointCanonicalization.ideal, coarseIdeal, rawSource, idealSide, ExactRecovery.delete] |
| exact @hdelete _ _ _ _ _ _ _ _ _ _ |
|
|
| @[simp] theorem sideVariable_zero {V : Type*} (x y z : V) : |
| sideVariable 0 x y z = x := rfl |
|
|
| @[simp] theorem sideVariable_one {V : Type*} (x y z : V) : |
| sideVariable 1 x y z = y := rfl |
|
|
| @[simp] theorem sideVariable_two {V : Type*} (x y z : V) : |
| sideVariable 2 x y z = z := rfl |
|
|
| theorem branch_eq_holedIdeal {I : Type*} (F : Type*) [CommRing F] |
| (allocation : Allocation) (m : ℕ) (ε : ℝ) |
| (a : JointExtraction.Assignment (Raw (K := K) (tick := tick) allocation m) |
| (Raw (K := K) (tick := tick) allocation m) (Raw (K := K) (tick := tick) allocation m) I) |
| (i : I) (e : Targets (K := K) (tick := tick) allocation m) |
| (hx : ∀ w, a.x w = some i → idealSide allocation m ε 0 e w) |
| (hy : ∀ w, a.y w = some i → idealSide allocation m ε 1 e w) |
| (hz : ∀ w, a.z w = some i → idealSide allocation m ε 2 e w) : |
| JointExtraction.branch |
| (ExactRecovery.delete (rawSource F allocation m) |
| (completeKeep allocation m ε 0) (completeKeep allocation m ε 1) |
| (completeKeep allocation m ε 2)) a i = |
| holedIdeal F allocation m ε e |
| (fun side w => sideVariable side (a.x w = some i) (a.y w = some i) (a.z w = some i)) := by |
| funext x y z |
| simp only [holedIdeal, sideVariable_zero, sideVariable_one, sideVariable_two] |
| by_cases ha : a.x x = some i ∧ a.y y = some i ∧ a.z z = some i |
| · have hix := hx x ha.1 |
| have hiy := hy y ha.2.1 |
| have hiz := hz z ha.2.2 |
| rw [JointExtraction.branch, ite_eq_left ha, ideal_eq_delete_source] |
| simp only [ExactRecovery.delete, hix, hiy, hiz, ha.1, ha.2.1, ha.2.2, true_and, ite_true] |
| · have hnot : ¬((a.x x = some i ∧ completeKeep allocation m ε 0 x) ∧ |
| (a.y y = some i ∧ completeKeep allocation m ε 1 y) ∧ |
| (a.z z = some i ∧ completeKeep allocation m ε 2 z)) := |
| fun h => ha ⟨h.1.1, h.2.1.1, h.2.2.1⟩ |
| rw [JointExtraction.branch, ite_eq_right ha] |
| simp only [ExactRecovery.delete, ite_eq_right hnot] |
|
|
| theorem support_filter_eq {F X Y Z : Type*} [Zero F] |
| [Fintype X] [Fintype Y] [Fintype Z] (Q : Tensor F X Y Z) : |
| (@Finset.filter (X × Y × Z) (fun p => Q p.1 p.2.1 p.2.2 ≠ 0) |
| (Classical.decPred _) Finset.univ) = RecoverySupport.support Q := by |
| ext p |
| simp only [RecoverySupport.support, Finset.mem_filter] |
|
|
| theorem repair_selected (F : Type*) [CommRing F] (allocation : Allocation) |
| {m : ℕ} {ε : ℝ} (hε : 0 < ε) |
| (hm : minimumDilation (K := K) (tick := tick) allocation ε ≤ m) |
| (e : Targets (K := K) (tick := tick) allocation m) |
| (assigned : Fin 3 → Raw (K := K) (tick := tick) allocation m → Prop) |
| (hcollision : |
| letI : MulAction (Symmetries (K := K) (tick := tick) allocation m) (Raw (K := K) (tick := tick) allocation m) := rawAction allocation m e |
| ∀ side w, Used F allocation m ε e side w → |
| (((OrbitCounting.orbitSet (G := Symmetries (K := K) (tick := tick) allocation m) w).filter |
| (fun v => completeKeep allocation m ε side v ∧ ¬assigned side v)).card : ℝ) / |
| (OrbitCounting.orbitSet (G := Symmetries (K := K) (tick := tick) allocation m) w).card ≤ 6 / m) : |
| InverseLinearRecovery.RecoveredBy (ideal F allocation m ε e) |
| (holedIdeal F allocation m ε e assigned) (shiftCount (K := K) (tick := tick) allocation ε m) := by |
| let _ : MulAction (Symmetries (K := K) (tick := tick) allocation m) (Raw (K := K) (tick := tick) allocation m) := rawAction allocation m e |
| have hm' := minimumDilation_spec allocation ε hm |
| have hc : 0 < AllFieldHistoryMasks.lossConstant (K := K) (tick := tick) allocation ε + 6 := by |
| have := AllFieldHistoryMasks.lossConstant_nonneg (K := K) (tick := tick) allocation ε |
| linarith |
| have hloss (side : Fin 3) (w : Raw (K := K) (tick := tick) allocation m) |
| (hw : Used F allocation m ε e side w) : |
| (((OrbitCounting.orbitSet (G := Symmetries (K := K) (tick := tick) allocation m) w).filter |
| (fun v => ¬(assigned side v ∧ completeKeep allocation m ε side v))).card : ℝ) / |
| (OrbitCounting.orbitSet (G := Symmetries (K := K) (tick := tick) allocation m) w).card ≤ |
| (AllFieldHistoryMasks.lossConstant (K := K) (tick := tick) allocation ε + 6) / m := by |
| have heq : ((OrbitCounting.orbitSet (G := Symmetries (K := K) (tick := tick) allocation m) w).filter |
| (fun v => ¬(assigned side v ∧ completeKeep allocation m ε side v))) = |
| ((OrbitCounting.orbitSet (G := Symmetries (K := K) (tick := tick) allocation m) w).filter |
| (fun v => ¬(completeKeep allocation m ε side v ∧ assigned side v))) := by |
| ext v |
| simp only [Finset.mem_filter] |
| exact and_congr Iff.rfl (not_congr and_comm) |
| rw [heq] |
| rw [OrbitLoss.missing_fraction_partition |
| (OrbitCounting.orbitSet (G := Symmetries (K := K) (tick := tick) allocation m) w) |
| (completeKeep allocation m ε side) (assigned side), add_div] |
| exact add_le_add (deterministic_loss_le F allocation hm'.1 hε e side w hw) |
| (hcollision side w hw) |
| unfold holedIdeal shiftCount |
| apply InverseLinearRecovery.repair_inverse_linear (G := Symmetries (K := K) (tick := tick) allocation m) |
| (ideal F allocation m ε e) |
| (fun w => assigned 0 w ∧ completeKeep allocation m ε 0 w) |
| (fun w => assigned 1 w ∧ completeKeep allocation m ε 1 w) |
| (fun w => assigned 2 w ∧ completeKeep allocation m ε 2 w) |
| (ideal_smul F allocation m ε e) (supportRate_pos (K := K) (tick := tick) allocation).le hc hm'.2 |
| · rw [support_filter_eq] |
| exact raw_support_card_le_exp allocation m (ideal F allocation m ε e) |
| · intro x hx |
| obtain ⟨y, z, hq⟩ := hx |
| exact hloss 0 x ⟨x, y, z, hq, rfl⟩ |
| · intro y hy |
| obtain ⟨x, z, hq⟩ := hy |
| exact hloss 1 y ⟨x, y, z, hq, rfl⟩ |
| · intro z hz |
| obtain ⟨x, y, hq⟩ := hz |
| exact hloss 2 z ⟨x, y, z, hq, rfl⟩ |
| |
| theorem repair_selected_branch {I : Type*} (F : Type*) [CommRing F] |
| (allocation : Allocation) {m : ℕ} {ε : ℝ} (hε : 0 < ε) |
| (hm : minimumDilation (K := K) (tick := tick) allocation ε ≤ m) |
| (a : JointExtraction.Assignment (Raw (K := K) (tick := tick) allocation m) |
| (Raw (K := K) (tick := tick) allocation m) (Raw (K := K) (tick := tick) allocation m) I) |
| (i : I) (e : Targets (K := K) (tick := tick) allocation m) |
| (hx : ∀ w, a.x w = some i → idealSide allocation m ε 0 e w) |
| (hy : ∀ w, a.y w = some i → idealSide allocation m ε 1 e w) |
| (hz : ∀ w, a.z w = some i → idealSide allocation m ε 2 e w) |
| (hcollision : |
| letI : MulAction (Symmetries (K := K) (tick := tick) allocation m) (Raw (K := K) (tick := tick) allocation m) := rawAction allocation m e |
| ∀ side w, Used F allocation m ε e side w → |
| (((OrbitCounting.orbitSet (G := Symmetries (K := K) (tick := tick) allocation m) w).filter |
| (fun v => completeKeep allocation m ε side v ∧ |
| ¬sideVariable side (a.x v = some i) (a.y v = some i) (a.z v = some i))).card : ℝ) / |
| (OrbitCounting.orbitSet (G := Symmetries (K := K) (tick := tick) allocation m) w).card ≤ 6 / m) : |
| InverseLinearRecovery.RecoveredBy (ideal F allocation m ε e) |
| (JointExtraction.branch |
| (ExactRecovery.delete (rawSource F allocation m) |
| (completeKeep allocation m ε 0) (completeKeep allocation m ε 1) |
| (completeKeep allocation m ε 2)) a i) (shiftCount (K := K) (tick := tick) allocation ε m) := by |
| rw [branch_eq_holedIdeal F allocation m ε a i e hx hy hz] |
| exact repair_selected F allocation hε hm e _ hcollision |
| |
| end MatrixMultiplication.AllFieldHistoryRecovery |
| |
| end |
| |
| end OAI |
| |
