File size: 16,061 Bytes
2f887db | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 | import OAI.Computability.PerfectCompleteness.Algebra.HierarchicalMatrixTable
import OAI.Computability.PerfectCompleteness.Decoding.LowerCutNodeCoordinates
import OAI.Computability.PerfectCompleteness.Decoding.LowerCutPairLemmas
import OAI.Computability.PerfectCompleteness.Decoding.LowerCutRowsLemmas
namespace OAI
section
namespace PerfectCompleteness.LowerCutPairBackground
open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays
open WholeCutGrouping
open UniqueGamesTheorem.Foundations.Games
open scoped Classical
abbrev F2 := ZMod 2
noncomputable section
variable {branch : Nat → Nat} {n m t : Nat}
(rows repeats : Nat → Nat) (p : Path branch n (m + 1))
(slots : Slots branch n → Fin t → MixedSupport.Slot)
abbrev Complement := LowerCutCallSplit.Complement rows repeats p slots
abbrev NativeRows := LowerCutPair.NativeRows rows p slots
abbrev NativeMatrix := LowerCutPair.NativeMatrix rows p slots
abbrev Observation := Complement rows repeats p slots ×
(NativeRows rows p slots × H (cutSlots p slots))
def splitObservation :
(CutChildGrouping.Assembled (C := LowerCutPair.Calls rows repeats p)
(cutSlots p slots) rows × H (cutSlots p slots)) →ₗ[F2]
Observation rows repeats p slots where
toFun z :=
((LowerCutCallSplit.splitAssembled rows repeats p slots z.1).1,
((LowerCutCallSplit.splitAssembled rows repeats p slots z.1).2, z.2))
map_add' _ _ := rfl
map_smul' _ _ := rfl
theorem splitObservation_surjective :
Function.Surjective (splitObservation rows repeats p slots) := by
rintro ⟨c, S, h⟩
obtain ⟨record, hrecord⟩ :=
LowerCutCallSplit.splitAssembled_surjective rows repeats p slots (c, S)
refine ⟨(record, h), ?_⟩
change ((LowerCutCallSplit.splitAssembled rows repeats p slots record).1,
((LowerCutCallSplit.splitAssembled rows repeats p slots record).2, h)) = (c, S, h)
rw [hrecord]
def observe (raw : LowerCutPair.Raw rows repeats p slots) :
Observation rows repeats p slots :=
splitObservation rows repeats p slots (CutCallExtension.observe (cutSlots p slots) rows raw)
theorem observe_law :
(CutChildGrouping.rawLaw (C := Option (LowerCutPair.Calls rows repeats p))
(cutSlots p slots) rows).pushforward (observe rows repeats p slots) =
(FiniteDistribution.uniform (Complement rows repeats p slots)).product
((FiniteDistribution.uniform (NativeRows rows p slots)).product
(FiniteDistribution.uniform (H (cutSlots p slots)))) := by
change (CutChildGrouping.rawLaw (C := Option (LowerCutPair.Calls rows repeats p))
(cutSlots p slots) rows).pushforward
(fun raw => splitObservation rows repeats p slots
(CutCallExtension.observe (cutSlots p slots) rows raw)) = _
rw [← FiniteDistribution.pushforward_comp, CutCallExtension.observe_law,
WholeCutSampler.uniform_product,
UniformLinearImage.uniform_pushforward_linearMap
(splitObservation rows repeats p slots) (splitObservation_surjective rows repeats p slots)]
rw [WholeCutSampler.uniform_product, WholeCutSampler.uniform_product]
@[simp] theorem observe_rows (exterior : Exterior rows repeats p slots)
(raw : LowerCutPair.Raw rows repeats p slots) :
(observe rows repeats p slots raw).2.1 =
SelectedArrayReplacement.selectedRows rows p slots
(LowerCutPair.first rows repeats p slots exterior raw) := by
exact (LowerCutRows.selectedRows_reconstruct_eq rows repeats p slots exterior _).symm
@[simp] theorem observe_fresh (raw : LowerCutPair.Raw rows repeats p slots) :
(observe rows repeats p slots raw).2.2 = LowerCutPair.fresh rows repeats p slots raw := rfl
def rowPair (exterior : Exterior rows repeats p slots)
(a : BucketSampler.Direction (rows (m + 1)))
(raw : LowerCutPair.Raw rows repeats p slots) :
Complement rows repeats p slots × (NativeRows rows p slots × NativeRows rows p slots) :=
((observe rows repeats p slots raw).1,
LowerCutPair.selectedPair rows repeats p slots exterior a raw)
theorem rowPair_law (exterior : Exterior rows repeats p slots)
(a : BucketSampler.Direction (rows (m + 1))) :
(CutChildGrouping.rawLaw (C := Option (LowerCutPair.Calls rows repeats p))
(cutSlots p slots) rows).pushforward (rowPair rows repeats p slots exterior a) =
(FiniteDistribution.uniform (Complement rows repeats p slots)).product
(LowerDirectionFiber.fiberPairLaw (H := H (cutSlots p slots)) a) := by
have h := congrArg
(fun μ : FiniteDistribution (Observation rows repeats p slots) => μ.pushforward
(fun z => (z.1, (z.2.1, LowerDirectionFiber.replace a z.2.1 z.2.2))))
(observe_law rows repeats p slots)
rw [FiniteDistribution.pushforward_comp] at h
have hp := FiniteDistribution.product_pushforward
(FiniteDistribution.uniform (Complement rows repeats p slots))
((FiniteDistribution.uniform (NativeRows rows p slots)).product
(FiniteDistribution.uniform (H (cutSlots p slots))))
(id : Complement rows repeats p slots → Complement rows repeats p slots)
(fun z => (z.1, LowerDirectionFiber.replace a z.1 z.2))
simp only [FiniteDistribution.pushforward_id, id_eq, LowerDirectionFiber.two_copy_law] at hp
have hfinal := h.trans hp
change (CutChildGrouping.rawLaw (C := Option (LowerCutPair.Calls rows repeats p))
(cutSlots p slots) rows).pushforward (fun raw =>
((observe rows repeats p slots raw).1,
(SelectedArrayReplacement.selectedRows rows p slots
(LowerCutPair.first rows repeats p slots exterior raw),
SelectedArrayReplacement.selectedRows rows p slots
(LowerCutPair.second rows repeats p slots exterior a raw)))) = _
simpa only [LowerCutPair.selected_second,
observe_rows rows repeats p slots exterior, observe_fresh] using hfinal
attribute [local instance] UniqueGamesTheorem.Appendix.RankLevelFilter.linearMapFintype
def matrixPair (exterior : Exterior rows repeats p slots)
(a : BucketSampler.Direction (rows (m + 1)))
(raw : LowerCutPair.Raw rows repeats p slots) :
Complement rows repeats p slots × (NativeMatrix rows p slots × NativeMatrix rows p slots) :=
((observe rows repeats p slots raw).1,
LowerCutPair.matrixPair rows repeats p slots exterior a raw)
theorem matrixPair_law (exterior : Exterior rows repeats p slots)
(a : BucketSampler.Direction (rows (m + 1))) :
(CutChildGrouping.rawLaw (C := Option (LowerCutPair.Calls rows repeats p))
(cutSlots p slots) rows).pushforward (matrixPair rows repeats p slots exterior a) =
(FiniteDistribution.uniform (Complement rows repeats p slots)).product
(((FiniteDistribution.uniform (NativeMatrix rows p slots)).product
(FiniteDistribution.uniform (H (cutSlots p slots)))).pushforward
(fun z => (z.1, EvaluationMatrix.shift (H (cutSlots p slots)) z.1 a.val z.2))) := by
have h := congrArg
(fun μ : FiniteDistribution
(Complement rows repeats p slots × (NativeRows rows p slots × NativeRows rows p slots)) =>
μ.pushforward (fun z => (z.1,
(EvaluationMatrix.ofRows (H (cutSlots p slots)) z.2.1,
EvaluationMatrix.ofRows (H (cutSlots p slots)) z.2.2))))
(rowPair_law rows repeats p slots exterior a)
rw [FiniteDistribution.pushforward_comp] at h
have hp := FiniteDistribution.product_pushforward
(FiniteDistribution.uniform (Complement rows repeats p slots))
(LowerDirectionFiber.fiberPairLaw (H := H (cutSlots p slots)) a)
(id : Complement rows repeats p slots → Complement rows repeats p slots)
(fun z => (EvaluationMatrix.ofRows (H (cutSlots p slots)) z.1,
EvaluationMatrix.ofRows (H (cutSlots p slots)) z.2))
simp only [FiniteDistribution.pushforward_id, id_eq, LowerDirectionFiber.matrix_pair_law] at hp
exact h.trans hp
def backgroundArrays (exterior : Exterior rows repeats p slots)
(c : Complement rows repeats p slots) : Arrays slots rows :=
LowerCutAssembled.reconstruct rows repeats p slots exterior
(LowerCutCallSplit.merge rows repeats p slots c 0)
def installRows (exterior : Exterior rows repeats p slots)
(c : Complement rows repeats p slots) (S : NativeRows rows p slots) : Arrays slots rows :=
SelectedArrayReplacement.replace rows p slots
(backgroundArrays rows repeats p slots exterior c) S
theorem reconstruct_merge (exterior : Exterior rows repeats p slots)
(c : Complement rows repeats p slots) (S : NativeRows rows p slots) :
LowerCutAssembled.reconstruct rows repeats p slots exterior
(LowerCutCallSplit.merge rows repeats p slots c S) =
installRows rows repeats p slots exterior c S := by
have h := LowerCutAssembled.reconstruct_congr_replace rows repeats p slots exterior
(LowerCutCallSplit.merge rows repeats p slots c 0)
(LowerCutCallSplit.merge rows repeats p slots c S)
(fun call hcall =>
(LowerCutCallSplit.merge_otherCall rows repeats p slots c S ⟨call, hcall⟩).trans
(LowerCutCallSplit.merge_otherCall rows repeats p slots c 0 ⟨call, hcall⟩).symm) rfl
rw [LowerCutCallSplit.selectRows_merge] at h
exact h
theorem first_eq_installRows (exterior : Exterior rows repeats p slots)
(raw : LowerCutPair.Raw rows repeats p slots) :
LowerCutPair.first rows repeats p slots exterior raw =
installRows rows repeats p slots exterior (observe rows repeats p slots raw).1
(SelectedArrayReplacement.selectedRows rows p slots
(LowerCutPair.first rows repeats p slots exterior raw)) := by
rw [← observe_rows rows repeats p slots exterior]
change LowerCutRows.reconstruct rows repeats p slots exterior
(CutCallExtension.original (cutSlots p slots) rows raw) =
installRows rows repeats p slots exterior
(LowerCutCallSplit.splitAssembled rows repeats p slots
(CutChildGrouping.assemble (cutSlots p slots) rows
(CutCallExtension.original (cutSlots p slots) rows raw))).1
(LowerCutCallSplit.splitAssembled rows repeats p slots
(CutChildGrouping.assemble (cutSlots p slots) rows
(CutCallExtension.original (cutSlots p slots) rows raw))).2
rw [← LowerCutAssembled.reconstruct_assemble, ← reconstruct_merge,
LowerCutCallSplit.merge_split]
private theorem replace_replace (arrays : Arrays slots rows)
(S T : NativeRows rows p slots) :
SelectedArrayReplacement.replace rows p slots
(SelectedArrayReplacement.replace rows p slots arrays S) T =
SelectedArrayReplacement.replace rows p slots arrays T := by
simp only [LowerCutNodeCoordinates.replace_eq_update]
funext node
by_cases hnode : node = WholeArrayInteriorExterior.upperNode p
· subst node
simp only [Function.update_self]
· simp only [Function.update_of_ne hnode]
theorem second_eq_installRows (exterior : Exterior rows repeats p slots)
(a : BucketSampler.Direction (rows (m + 1)))
(raw : LowerCutPair.Raw rows repeats p slots) :
LowerCutPair.second rows repeats p slots exterior a raw =
installRows rows repeats p slots exterior (observe rows repeats p slots raw).1
(SelectedArrayReplacement.selectedRows rows p slots
(LowerCutPair.second rows repeats p slots exterior a raw)) := by
rw [LowerCutPair.selected_second]
exact (congrArg
(fun arrays => SelectedArrayReplacement.replace rows p slots arrays
(LowerDirectionFiber.replace a
(SelectedArrayReplacement.selectedRows rows p slots
(LowerCutPair.first rows repeats p slots exterior raw))
(LowerCutPair.fresh rows repeats p slots raw)))
(first_eq_installRows rows repeats p slots exterior raw)).trans
(replace_replace rows p slots _ _ _)
def installMatrix (exterior : Exterior rows repeats p slots)
(c : Complement rows repeats p slots) (X : NativeMatrix rows p slots) : Arrays slots rows :=
installRows rows repeats p slots exterior c (EvaluationMatrix.rows (H (cutSlots p slots)) X)
theorem arraysPair_eq (exterior : Exterior rows repeats p slots)
(a : BucketSampler.Direction (rows (m + 1)))
(raw : LowerCutPair.Raw rows repeats p slots) :
LowerCutPair.arraysPair rows repeats p slots exterior a raw =
(installMatrix rows repeats p slots exterior (matrixPair rows repeats p slots exterior a raw).1
(matrixPair rows repeats p slots exterior a raw).2.1,
installMatrix rows repeats p slots exterior (matrixPair rows repeats p slots exterior a raw).1
(matrixPair rows repeats p slots exterior a raw).2.2) := by
simp only [matrixPair, LowerCutPair.matrixPair, LowerCutPair.selectedPair,
installMatrix, EvaluationMatrix.rows_ofRows]
exact Prod.ext (first_eq_installRows rows repeats p slots exterior raw)
(second_eq_installRows rows repeats p slots exterior a raw)
theorem arraysPair_law (exterior : Exterior rows repeats p slots)
(a : BucketSampler.Direction (rows (m + 1))) :
(CutChildGrouping.rawLaw (C := Option (LowerCutPair.Calls rows repeats p))
(cutSlots p slots) rows).pushforward (LowerCutPair.arraysPair rows repeats p slots exterior a) =
((FiniteDistribution.uniform (Complement rows repeats p slots)).product
(((FiniteDistribution.uniform (NativeMatrix rows p slots)).product
(FiniteDistribution.uniform (H (cutSlots p slots)))).pushforward
(fun z => (z.1, EvaluationMatrix.shift (H (cutSlots p slots)) z.1 a.val z.2)))).pushforward
(fun z => (installMatrix rows repeats p slots exterior z.1 z.2.1,
installMatrix rows repeats p slots exterior z.1 z.2.2)) := by
have h := congrArg
(fun μ : FiniteDistribution
(Complement rows repeats p slots × (NativeMatrix rows p slots × NativeMatrix rows p slots)) =>
μ.pushforward (fun z => (installMatrix rows repeats p slots exterior z.1 z.2.1,
installMatrix rows repeats p slots exterior z.1 z.2.2)))
(matrixPair_law rows repeats p slots exterior a)
rw [FiniteDistribution.pushforward_comp] at h
simpa only [← arraysPair_eq] using h
theorem installMatrix_eq_update (exterior : Exterior rows repeats p slots)
(c : Complement rows repeats p slots) (X : NativeMatrix rows p slots) :
installMatrix rows repeats p slots exterior c X =
Function.update (backgroundArrays rows repeats p slots exterior c)
(WholeArrayInteriorExterior.upperNode p)
(LowerCutNodeCoordinates.rowEquiv rows p slots
(EvaluationMatrix.rows (H (cutSlots p slots)) X)) :=
LowerCutNodeCoordinates.replace_eq_update rows p slots _ _
theorem installMatrix_outside (exterior : Exterior rows repeats p slots)
(c : Complement rows repeats p slots) (X : NativeMatrix rows p slots)
(node : Nodes branch n) (hne : node ≠ WholeArrayInteriorExterior.upperNode p) :
installMatrix rows repeats p slots exterior c X node =
backgroundArrays rows repeats p slots exterior c node :=
SelectedArrayReplacement.replace_outside rows p slots _ _ node hne
theorem backgroundOf_installMatrix (exterior : Exterior rows repeats p slots)
(c : Complement rows repeats p slots) (X : NativeMatrix rows p slots) :
HierarchicalMatrixTable.backgroundOf slots (WholeArrayInteriorExterior.upperNode p)
(installMatrix rows repeats p slots exterior c X) =
HierarchicalMatrixTable.backgroundOf slots (WholeArrayInteriorExterior.upperNode p)
(backgroundArrays rows repeats p slots exterior c) := by
funext node
exact installMatrix_outside rows repeats p slots exterior c X node.val node.property
theorem upperMatrix_installMatrix (exterior : Exterior rows repeats p slots)
(c : Complement rows repeats p slots) (X : NativeMatrix rows p slots)
(upper : Nodes branch n) (hne : upper ≠ WholeArrayInteriorExterior.upperNode p) :
NodeEmbedding.matrix (installMatrix rows repeats p slots exterior c X) upper =
NodeEmbedding.matrix (backgroundArrays rows repeats p slots exterior c) upper := by
unfold NodeEmbedding.matrix NodeEmbedding.embeddedRows
rw [installMatrix_outside rows repeats p slots exterior c X upper hne]
end
end PerfectCompleteness.LowerCutPairBackground
end
end OAI
|