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cdec422 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 | import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.GroupCore
import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.Recovery
import OAI.LinearAlgebra.MatrixMultiplication.Recovery.GroupOrbitProjection
import OAI.LinearAlgebra.MatrixMultiplication.Recovery.RecoveryOrbitDecisions
namespace OAI
/-! Finite extraction histories, inherited masks and recovery bounds. -/
noncomputable section
namespace MatrixMultiplication.AllFieldHistoryGroupedRecovery
open AllFieldHistory AllFieldHistorySupport AllFieldHistoryChildLaws
open AllFieldHistoryGroupMasks JointPopulation JointCanonicalization JointCanonicalCW
open PermutationMatching
open scoped BigOperators
attribute [local instance] Classical.propDecidable Classical.decEq
variable {K tick : ℕ}
theorem coordinates_project (allocation : Allocation) (m : ℕ) (sigma : Placement)
(e : AllFieldHistoryRecovery.Targets (K := K) (tick := tick) allocation m)
(w : AllFieldHistoryRecovery.Raw (K := K) (tick := tick) allocation m) :
GroupOrbitProjection.projectPair
(ClassPositions (activeCounts allocation m))
AllFieldHistoryMasks.Letters AllFieldHistoryMasks.Letters
(fun h : Active K tick => h.val.physicalOrder = sigma)
(AllFieldHistoryRecovery.coordinates allocation m e w) =
groupCoordinates allocation m sigma (projectTarget allocation m sigma e)
(projectRaw allocation m sigma w) := rfl
theorem idealSide_iff_all_groups (allocation : Allocation) (m : ℕ) (ε : ℝ)
(side : Fin 3)
(e : AllFieldHistoryRecovery.Targets (K := K) (tick := tick) allocation m)
(w : AllFieldHistoryRecovery.Raw (K := K) (tick := tick) allocation m) :
AllFieldHistoryRecovery.idealSide allocation m ε side e w ↔
∀ sigma, groupIdealSide allocation m ε side sigma
(projectTarget allocation m sigma e) (projectRaw allocation m sigma w) := by
constructor
· intro hw sigma
exact ⟨fun h j => hw.1 h.val j, fun h u hu => hw.2 h.val u hu⟩
· intro hw
exact ⟨fun h j => (hw h.val.physicalOrder).1 ⟨h, rfl⟩ j,
fun h u hu => (hw h.val.physicalOrder).2 ⟨h, rfl⟩ u hu⟩
theorem project_joinGroupTargets (allocation : Allocation) (m : ℕ)
(e : ∀ sigma, GroupTarget (K := K) (tick := tick) allocation m sigma)
(sigma : Placement) :
projectTarget allocation m sigma (joinGroupTargets (activeCounts allocation m) e) =
e sigma := by
funext h
rcases h with ⟨h, hh⟩
cases hh
rfl
def groupedAssignment (allocation : Allocation) (m : ℕ) (E : Placement → Type*)
(a : ∀ sigma, JointExtraction.Assignment
(GroupRaw (K := K) (tick := tick) allocation m sigma)
(GroupRaw (K := K) (tick := tick) allocation m sigma) (GroupRaw (K := K) (tick := tick) allocation m sigma) (E sigma)) :
JointExtraction.Assignment (AllFieldHistoryRecovery.Raw (K := K) (tick := tick) allocation m)
(AllFieldHistoryRecovery.Raw (K := K) (tick := tick) allocation m) (AllFieldHistoryRecovery.Raw (K := K) (tick := tick) allocation m)
(∀ sigma, E sigma) where
x w := GroupAssignment.collect (fun sigma => (a sigma).x)
(fun sigma => projectRaw allocation m sigma w)
y w := GroupAssignment.collect (fun sigma => (a sigma).y)
(fun sigma => projectRaw allocation m sigma w)
z w := GroupAssignment.collect (fun sigma => (a sigma).z)
(fun sigma => projectRaw allocation m sigma w)
theorem groupedAssignment_sound (allocation : Allocation) (m : ℕ) (ε : ℝ)
(E : Placement → Type*)
(a : ∀ sigma, JointExtraction.Assignment
(GroupRaw (K := K) (tick := tick) allocation m sigma)
(GroupRaw (K := K) (tick := tick) allocation m sigma) (GroupRaw (K := K) (tick := tick) allocation m sigma) (E sigma))
(i : ∀ sigma, E sigma) (e : ∀ sigma, GroupTarget (K := K) (tick := tick) allocation m sigma)
(hx : ∀ sigma w, (a sigma).x w = some (i sigma) →
groupIdealSide allocation m ε 0 sigma (e sigma) w)
(hy : ∀ sigma w, (a sigma).y w = some (i sigma) →
groupIdealSide allocation m ε 1 sigma (e sigma) w)
(hz : ∀ sigma w, (a sigma).z w = some (i sigma) →
groupIdealSide allocation m ε 2 sigma (e sigma) w) :
(∀ w, (groupedAssignment allocation m E a).x w = some i →
AllFieldHistoryRecovery.idealSide allocation m ε 0
(joinGroupTargets (activeCounts allocation m) e) w) ∧
(∀ w, (groupedAssignment allocation m E a).y w = some i →
AllFieldHistoryRecovery.idealSide allocation m ε 1
(joinGroupTargets (activeCounts allocation m) e) w) ∧
(∀ w, (groupedAssignment allocation m E a).z w = some i →
AllFieldHistoryRecovery.idealSide allocation m ε 2
(joinGroupTargets (activeCounts allocation m) e) w) := by
refine ⟨?_, ?_, ?_⟩
· intro w hw
apply (idealSide_iff_all_groups allocation m ε 0 _ w).2
intro sigma
rw [project_joinGroupTargets]
exact hx sigma _ ((GroupAssignment.collect_eq_some _ _ _).mp hw sigma)
· intro w hw
apply (idealSide_iff_all_groups allocation m ε 1 _ w).2
intro sigma
rw [project_joinGroupTargets]
exact hy sigma _ ((GroupAssignment.collect_eq_some _ _ _).mp hw sigma)
· intro w hw
apply (idealSide_iff_all_groups allocation m ε 2 _ w).2
intro sigma
rw [project_joinGroupTargets]
exact hz sigma _ ((GroupAssignment.collect_eq_some _ _ _).mp hw sigma)
def localOrbit (allocation : Allocation) (m : ℕ) (sigma : Placement)
(e : GroupTarget (K := K) (tick := tick) allocation m sigma)
(w : GroupRaw (K := K) (tick := tick) allocation m sigma) : Finset (GroupRaw (K := K) (tick := tick) allocation m sigma) :=
GroupOrbitProjection.rawOrbit (groupCounts allocation m sigma)
(fun h => Fin (activeHalfLength h.val) → Fin 7)
(fun h => Fin (activeHalfLength h.val) → Fin 7) e w
def localAssigned (allocation : Allocation) (m : ℕ) (sigma : Placement) {E : Type*}
(a : JointExtraction.Assignment
(GroupRaw (K := K) (tick := tick) allocation m sigma)
(GroupRaw (K := K) (tick := tick) allocation m sigma) (GroupRaw (K := K) (tick := tick) allocation m sigma) E)
(i : E) (side : Fin 3) (w : GroupRaw (K := K) (tick := tick) allocation m sigma) : Prop :=
AllFieldHistoryRecovery.sideVariable side (a.x w = some i) (a.y w = some i) (a.z w = some i)
theorem grouped_assigned_iff (allocation : Allocation) (m : ℕ)
(E : Placement → Type*)
(a : ∀ sigma, JointExtraction.Assignment
(GroupRaw (K := K) (tick := tick) allocation m sigma)
(GroupRaw (K := K) (tick := tick) allocation m sigma) (GroupRaw (K := K) (tick := tick) allocation m sigma) (E sigma))
(i : ∀ sigma, E sigma) (side : Fin 3)
(w : AllFieldHistoryRecovery.Raw (K := K) (tick := tick) allocation m) :
AllFieldHistoryRecovery.sideVariable side
((groupedAssignment allocation m E a).x w = some i)
((groupedAssignment allocation m E a).y w = some i)
((groupedAssignment allocation m E a).z w = some i) ↔
∀ sigma, localAssigned allocation m sigma (a sigma) (i sigma) side
(projectRaw allocation m sigma w) := by
fin_cases side <;> exact GroupAssignment.collect_eq_some _ _ _
theorem filter_eq_rejected {V : Type*} (orbit : Finset V)
(bad : V → Prop) (d : DecidablePred bad) :
@Finset.filter V bad d orbit = rejectedFilter orbit bad := by
ext v
simp only [rejectedFilter, Finset.mem_filter]
theorem grouped_collision_le (allocation : Allocation) (m : ℕ) (ε : ℝ)
(E : Placement → Type*)
(a : ∀ sigma, JointExtraction.Assignment
(GroupRaw (K := K) (tick := tick) allocation m sigma)
(GroupRaw (K := K) (tick := tick) allocation m sigma) (GroupRaw (K := K) (tick := tick) allocation m sigma) (E sigma))
(i : ∀ sigma, E sigma) (e : ∀ sigma, GroupTarget (K := K) (tick := tick) allocation m sigma)
(side : Fin 3) (w : AllFieldHistoryRecovery.Raw (K := K) (tick := tick) allocation m)
(hbound : ∀ sigma,
(((localOrbit allocation m sigma (e sigma) (projectRaw allocation m sigma w)).filter
(fun v => AllFieldHistoryGroupMasks.completeKeep allocation m ε side sigma v ∧
¬localAssigned allocation m sigma (a sigma) (i sigma) side v)).card : ℝ) /
(localOrbit allocation m sigma (e sigma) (projectRaw allocation m sigma w)).card ≤ 1 / m) :
letI : MulAction (AllFieldHistoryRecovery.Symmetries (K := K) (tick := tick) allocation m)
(AllFieldHistoryRecovery.Raw (K := K) (tick := tick) allocation m) :=
AllFieldHistoryRecovery.rawAction allocation m (joinGroupTargets (activeCounts allocation m) e)
(((OrbitCounting.orbitSet (G := AllFieldHistoryRecovery.Symmetries (K := K) (tick := tick) allocation m) w).filter
(fun v => AllFieldHistoryRecovery.completeKeep allocation m ε side v ∧
¬AllFieldHistoryRecovery.sideVariable side
((groupedAssignment allocation m E a).x v = some i)
((groupedAssignment allocation m E a).y v = some i)
((groupedAssignment allocation m E a).z v = some i))).card : ℝ) /
(OrbitCounting.orbitSet (G := AllFieldHistoryRecovery.Symmetries (K := K) (tick := tick) allocation m) w).card ≤ 6 / m := by
let target := joinGroupTargets (activeCounts allocation m) e
let : MulAction (AllFieldHistoryRecovery.Symmetries (K := K) (tick := tick) allocation m)
(AllFieldHistoryRecovery.Raw (K := K) (tick := tick) allocation m) := AllFieldHistoryRecovery.rawAction allocation m target
have hb (sigma : Placement) :
((rejectedFilter (GroupOrbitProjection.rawOrbit
(GroupOrbitProjection.groupCounts (activeCounts allocation m)
(fun h : Active K tick => h.val.physicalOrder = sigma))
(fun h => Left activeHalfLength h.val) (fun h => Right activeHalfLength h.val)
(GroupOrbitProjection.projectTarget (activeCounts allocation m)
(fun h => h.val.physicalOrder = sigma) target)
(projectRaw allocation m sigma w))
(fun v => AllFieldHistoryGroupMasks.completeKeep allocation m ε side sigma v ∧
¬localAssigned allocation m sigma (a sigma) (i sigma) side v)).card : ℝ) /
(GroupOrbitProjection.rawOrbit
(GroupOrbitProjection.groupCounts (activeCounts allocation m)
(fun h : Active K tick => h.val.physicalOrder = sigma))
(fun h => Left activeHalfLength h.val) (fun h => Right activeHalfLength h.val)
(GroupOrbitProjection.projectTarget (activeCounts allocation m)
(fun h => h.val.physicalOrder = sigma) target)
(projectRaw allocation m sigma w)).card ≤ 1 / m := by
change ((rejectedFilter
(localOrbit allocation m sigma (projectTarget allocation m sigma target)
(projectRaw allocation m sigma w))
(fun v => AllFieldHistoryGroupMasks.completeKeep allocation m ε side sigma v ∧
¬localAssigned allocation m sigma (a sigma) (i sigma) side v)).card : ℝ) /
(localOrbit allocation m sigma (projectTarget allocation m sigma target)
(projectRaw allocation m sigma w)).card ≤ 1 / m
dsimp only [target]
rw [project_joinGroupTargets]
have hb0 := hbound sigma
rw [filter_eq_rejected] at hb0
exact hb0
have hh := GroupOrbitProjection.raw_group_passing_rejection_fraction_le
(activeCounts allocation m) (Left activeHalfLength) (Right activeHalfLength)
(fun h : Active K tick => h.val.physicalOrder) target w
(AllFieldHistoryRecovery.completeKeep allocation m ε side)
(fun sigma => AllFieldHistoryGroupMasks.completeKeep allocation m ε side sigma)
(fun sigma => localAssigned allocation m sigma (a sigma) (i sigma) side)
(fun v hv sigma => completeKeep_project allocation m ε side v hv sigma)
(fun _ => (1 : ℝ) / m) hb
have hcard : Fintype.card Placement = 6 := by
norm_num [Placement, Fintype.card_perm, Nat.factorial]
have horbit :
OrbitCounting.orbitSet
(G := AllFieldHistoryRecovery.Symmetries (K := K) (tick := tick) allocation m) w =
GroupOrbitProjection.rawOrbit (activeCounts allocation m)
(Left activeHalfLength) (Right activeHalfLength) target w := by
exact RecoveryOrbitDecisions.orbitSet_eq_transportedOrbit
(G := AllFieldHistoryRecovery.Symmetries (K := K) (tick := tick) allocation m)
_ _ _ (AllFieldHistoryRecovery.coordinates allocation m target) w
rw [filter_eq_rejected, horbit]
have hfilter := rejectedFilter_congr
(GroupOrbitProjection.rawOrbit (activeCounts allocation m)
(Left activeHalfLength) (Right activeHalfLength) target w)
(fun v => AllFieldHistoryRecovery.completeKeep allocation m ε side v ∧
¬AllFieldHistoryRecovery.sideVariable side
((groupedAssignment allocation m E a).x v = some i)
((groupedAssignment allocation m E a).y v = some i)
((groupedAssignment allocation m E a).z v = some i))
(fun v => AllFieldHistoryRecovery.completeKeep allocation m ε side v ∧
¬∀ sigma, localAssigned allocation m sigma (a sigma) (i sigma) side
(GroupOrbitProjection.projectRaw (activeCounts allocation m)
(Left activeHalfLength) (Right activeHalfLength)
(fun h : Active K tick => h.val.physicalOrder = sigma) v))
(fun v => and_congr Iff.rfl
(not_congr (grouped_assigned_iff allocation m E a i side v)))
rw [hfilter]
simpa only [Finset.sum_const, Finset.card_univ, hcard, nsmul_eq_mul, Nat.cast_ofNat, mul_one_div] using hh
theorem repair_grouped (F : Type*) [CommRing F] (allocation : Allocation)
{m : ℕ} {ε : ℝ} (hε : 0 < ε)
(hm : AllFieldHistoryRecovery.minimumDilation (K := K) (tick := tick) allocation ε ≤ m)
(E : Placement → Type*)
(a : ∀ sigma, JointExtraction.Assignment
(GroupRaw (K := K) (tick := tick) allocation m sigma)
(GroupRaw (K := K) (tick := tick) allocation m sigma) (GroupRaw (K := K) (tick := tick) allocation m sigma) (E sigma))
(i : ∀ sigma, E sigma) (e : ∀ sigma, GroupTarget (K := K) (tick := tick) allocation m sigma)
(hx : ∀ sigma w, (a sigma).x w = some (i sigma) →
groupIdealSide allocation m ε 0 sigma (e sigma) w)
(hy : ∀ sigma w, (a sigma).y w = some (i sigma) →
groupIdealSide allocation m ε 1 sigma (e sigma) w)
(hz : ∀ sigma w, (a sigma).z w = some (i sigma) →
groupIdealSide allocation m ε 2 sigma (e sigma) w)
(hbound : ∀ side w,
AllFieldHistoryRecovery.Used F allocation m ε
(joinGroupTargets (activeCounts allocation m) e) side w → ∀ sigma,
(((localOrbit allocation m sigma (e sigma) (projectRaw allocation m sigma w)).filter
(fun v => AllFieldHistoryGroupMasks.completeKeep allocation m ε side sigma v ∧
¬localAssigned allocation m sigma (a sigma) (i sigma) side v)).card : ℝ) /
(localOrbit allocation m sigma (e sigma) (projectRaw allocation m sigma w)).card ≤ 1 / m) :
InverseLinearRecovery.RecoveredBy
(AllFieldHistoryRecovery.ideal F allocation m ε (joinGroupTargets (activeCounts allocation m) e))
(JointExtraction.branch
(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))
(groupedAssignment allocation m E a) i)
(AllFieldHistoryRecovery.shiftCount (K := K) (tick := tick) allocation ε m) := by
have hs := groupedAssignment_sound allocation m ε E a i e hx hy hz
apply AllFieldHistoryRecovery.repair_selected_branch F allocation hε hm
(groupedAssignment allocation m E a) i (joinGroupTargets (activeCounts allocation m) e)
hs.1 hs.2.1 hs.2.2
intro side w hw
exact grouped_collision_le allocation m ε E a i e side w (hbound side w hw)
end MatrixMultiplication.AllFieldHistoryGroupedRecovery
end
end OAI
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