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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
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