File size: 16,012 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 | import OAI.Computability.PerfectCompleteness.Construction.SourceChildKernelLemmas
import OAI.Computability.PerfectCompleteness.Repetition.CleanEndpointSlots
namespace OAI
section
namespace PerfectCompleteness.CleanRecordChildBlocks
noncomputable section
open scoped Classical
open UniqueGamesTheorem.Foundations.Games
open RecursiveSpaces TreeSourceSpaces HierarchicalArrays
open SourceQuestionReconstruction
variable {branch : Nat → Nat} {h t v m : Nat} {C : Type*} [Fintype C]
(rows : Nat → Nat) (clauses : Fin m → SourceClause.NormalizedClause v)
(designated : Fin (branch h) → Slots branch h)
abbrev Record :=
CleanSourceRecord.Record
(D := SourceChildKernel.D (C := C) (t := t) rows clauses designated) designated
(SourceChildKernel.Factors (C := C) (t := t) rows clauses designated)
abbrev Sample := SourceChildKernel.Sample (C := C) (t := t) rows clauses designated
abbrev Row (i : Fin (branch h)) := SourceTuple m t ×
(SourceChildKernel.Rest (m := m) (t := t) designated i ×
SourceChildKernel.Raw (C := C) (t := t) rows clauses designated i)
def clean (record : Record (C := C) (t := t) rows clauses designated) :
Fin (branch h) → Prop :=
CleanSourceRecord.clean designated
(SourceChildKernel.Factors (C := C) (t := t) rows clauses designated) record
def visible (record : Record (C := C) (t := t) rows clauses designated) :
Visible (E := SourceTuple m t) designated (clean rows clauses designated record) :=
CleanSourceRecord.visible designated
(SourceChildKernel.Factors (C := C) (t := t) rows clauses designated) record
def projected (record : Record (C := C) (t := t) rows clauses designated) :
Fin (branch h) → Bool :=
CleanSourceRecord.projected designated
(SourceChildKernel.Factors (C := C) (t := t) rows clauses designated) record
def event (record : Record (C := C) (t := t) rows clauses designated) :
Sample (C := C) (t := t) rows clauses designated → Bool :=
UniformCleanSources.recordEvent
(SourceChildKernel.Factors (C := C) (t := t) rows clauses designated)
(SourceChildKernel.zero (C := C) (t := t) rows clauses designated) record
abbrev Nonclean (record : Record (C := C) (t := t) rows clauses designated) :=
{i : Fin (branch h) // ¬ clean rows clauses designated record i}
def extract (record : Record (C := C) (t := t) rows clauses designated)
(i : Nonclean rows clauses designated record) : Row (C := C) (t := t) rows clauses designated i.val :=
match hr : record i.val with
| .inl row => row
| .inr _ => False.elim (i.property (by
simp only [clean, CleanSourceRecord.clean, hr, UniformCleanSources.cleanRecord]))
theorem record_extract (record : Record (C := C) (t := t) rows clauses designated)
(i : Nonclean rows clauses designated record) :
record i.val = .inl (extract rows clauses designated record i) := by
unfold extract
split
· rename_i row hr
exact hr
· rename_i rest hr
exact False.elim (i.property (by
simp only [clean, CleanSourceRecord.clean, hr, UniformCleanSources.cleanRecord]))
theorem extract_eq (record : Record (C := C) (t := t) rows clauses designated)
(x : Sample (C := C) (t := t) rows clauses designated)
(hevent : event rows clauses designated record x = true)
(i : Nonclean rows clauses designated record) :
extract rows clauses designated record i = x i.val := by
have obs := CleanSourceRecord.observation_eq designated
(SourceChildKernel.Factors (C := C) (t := t) rows clauses designated)
(SourceChildKernel.zero (C := C) (t := t) rows clauses designated) record x hevent i.val
cases hc : UniformCleanSources.childClean
(SourceChildKernel.Factors (C := C) (t := t) rows clauses designated)
(SourceChildKernel.zero (C := C) (t := t) rows clauses designated)
i.val (x i.val).1 (x i.val).2.1 (x i.val).2.2 with
| false =>
have hr : record i.val = .inl (x i.val) := by
simpa only [SelectedSources.childObservation, hc, Bool.false_eq_true, ite_false]
using obs.symm
exact Sum.inl.inj ((record_extract rows clauses designated record i).symm.trans hr)
| true =>
have hr : record i.val = .inr (x i.val).2.1 := by
simpa only [SelectedSources.childObservation, hc, ite_true] using obs.symm
exact False.elim (i.property (by
simp only [clean, CleanSourceRecord.clean, hr, UniformCleanSources.cleanRecord]))
def source (record : Record (C := C) (t := t) rows clauses designated)
(i : Nonclean rows clauses designated record) :
SourceChildKernel.Source (m := m) (t := t) designated i.val :=
((extract rows clauses designated record i).1, (extract rows clauses designated record i).2.1)
def raw (record : Record (C := C) (t := t) rows clauses designated)
(i : Nonclean rows clauses designated record) :
SourceChildKernel.Raw (C := C) (t := t) rows clauses designated i.val :=
(extract rows clauses designated record i).2.2
theorem visible_nonclean (record : Record (C := C) (t := t) rows clauses designated)
(i : Nonclean rows clauses designated record) (leaf : Slots branch h)
(hleaf : ¬ (clean rows clauses designated record i.val ∧ leaf = designated i.val)) :
visible rows clauses designated record i.val ⟨leaf, hleaf⟩ =
SourceChildKernel.tree designated i.val (source rows clauses designated record i) leaf := by
dsimp only [visible, CleanSourceRecord.visible]
split
· rename_i row hr
have hrow : row = extract rows clauses designated record i :=
Sum.inl.inj (hr.symm.trans (record_extract rows clauses designated record i))
rw [hrow]
rfl
· rename_i rest hr
have hbad := hr.symm.trans (record_extract rows clauses designated record i)
cases hbad
theorem projected_nonclean (record : Record (C := C) (t := t) rows clauses designated)
(i : Nonclean rows clauses designated record) :
projected rows clauses designated record i.val =
SourceChildKernel.rawProjected rows clauses designated i.val
(raw rows clauses designated record i) := by
simp only [projected, CleanSourceRecord.projected,
record_extract rows clauses designated record i, raw,
CleanSourceRecord.rawProjected, SourceChildKernel.rawProjected]
cases (extract rows clauses designated record i).2.2 <;> rfl
variable [NeZero m]
def leftRef (record : Record (C := C) (t := t) rows clauses designated) :
Slots branch (h + 1) → Fin t → MixedSupport.Slot :=
CleanEndpointSlots.leftReference clauses designated (clean rows clauses designated record)
(visible rows clauses designated record)
def rightRef (record : Record (C := C) (t := t) rows clauses designated) :
Slots branch (h + 1) → Fin t → MixedSupport.Slot :=
CleanEndpointSlots.rightReference clauses designated (clean rows clauses designated record)
(visible rows clauses designated record) (projected rows clauses designated record)
theorem leftRef_childSlots (record : Record (C := C) (t := t) rows clauses designated)
(i : Nonclean rows clauses designated record) :
childSlots (leftRef rows clauses designated record) i.val =
SourceChildKernel.nativeSlots clauses designated i.val (source rows clauses designated record i) := by
funext leaf k
have hn : ¬ (clean rows clauses designated record i.val ∧ leaf = designated i.val) :=
fun h => i.property h.1
simp only [leftRef, CleanEndpointSlots.leftReference, CleanEndpointSlots.leftInside,
childSlots, reconstructLeft, reconstruct, dite_eq_right hn]
rw [visible_nonclean rows clauses designated record i leaf hn]
rfl
theorem rightRef_childSlots (record : Record (C := C) (t := t) rows clauses designated)
(i : Nonclean rows clauses designated record) :
childSlots (rightRef rows clauses designated record) i.val =
SourceChildKernel.mixedSlots clauses designated i.val (source rows clauses designated record i)
(SourceChildKernel.rawProjected rows clauses designated i.val (raw rows clauses designated record i)) := by
funext leaf k
have hn : ¬ (clean rows clauses designated record i.val ∧ leaf = designated i.val) :=
fun h => i.property h.1
simp only [rightRef, CleanEndpointSlots.rightReference, CleanEndpointSlots.rightInside,
childSlots, reconstructRight, reconstruct, dite_eq_right hn]
rw [visible_nonclean rows clauses designated record i leaf hn,
projected_nonclean rows clauses designated record i]
exact (SourceChildKernel.mixedSlots_eq_rightTuple clauses designated i.val
(source rows clauses designated record i)
(SourceChildKernel.rawProjected rows clauses designated i.val (raw rows clauses designated record i))
leaf k).symm
theorem leftRef_childSlots_event (record : Record (C := C) (t := t) rows clauses designated)
(x : Sample (C := C) (t := t) rows clauses designated)
(hevent : event rows clauses designated record x = true)
(i : Nonclean rows clauses designated record) :
childSlots (leftRef rows clauses designated record) i.val =
childSlots (SourceChildKernel.parentLeftSlots clauses designated
(fun j => ((x j).1, (x j).2.1))) i.val :=
(leftRef_childSlots rows clauses designated record i).trans
(congrArg (fun y : Row (C := C) (t := t) rows clauses designated i.val =>
SourceChildKernel.nativeSlots clauses designated i.val (y.1, y.2.1))
(extract_eq rows clauses designated record x hevent i))
theorem rightRef_childSlots_event (record : Record (C := C) (t := t) rows clauses designated)
(x : Sample (C := C) (t := t) rows clauses designated)
(hevent : event rows clauses designated record x = true)
(i : Nonclean rows clauses designated record) :
childSlots (rightRef rows clauses designated record) i.val =
childSlots (SourceChildKernel.parentRightSlots rows clauses designated
(fun j => ((x j).1, (x j).2.1)) (fun j => (x j).2.2)) i.val :=
(rightRef_childSlots rows clauses designated record i).trans
(congrArg (fun y : Row (C := C) (t := t) rows clauses designated i.val =>
SourceChildKernel.mixedSlots clauses designated i.val (y.1, y.2.1)
(SourceChildKernel.rawProjected rows clauses designated i.val y.2.2))
(extract_eq rows clauses designated record x hevent i))
def physicalRight (i : Fin (branch h)) (row : Row (C := C) (t := t) rows clauses designated i) :
SourceChildKernel.Block C rows (SourceChildKernel.mixedSlots clauses designated i (row.1, row.2.1)
(SourceChildKernel.rawProjected rows clauses designated i row.2.2)) :=
SourceChildKernel.decodeRight rows clauses designated i (row.1, row.2.1) row.2.2
def physicalLeft (i : Fin (branch h)) (row : Row (C := C) (t := t) rows clauses designated i) :
SourceChildKernel.Block C rows (SourceChildKernel.nativeSlots clauses designated i (row.1, row.2.1)) :=
let p := SourceChildKernel.projection clauses designated i (row.1, row.2.1)
(SourceChildKernel.rawProjected rows clauses designated i row.2.2)
let block := physicalRight rows clauses designated i row
(fun call => ChildBlockProjection.squarePullback p (block.1 call),
ChildBlockProjection.arraysPullback rows p block.2)
omit [NeZero m] in
theorem physicalLeft_eq_leftChildren
(x : Sample (C := C) (t := t) rows clauses designated) (i : Fin (branch h)) :
physicalLeft rows clauses designated i (x i) =
SourceChildKernel.leftChildren rows clauses designated
(fun j => ((x j).1, (x j).2.1)) (fun j => (x j).2.2) i := by
rw [SourceChildKernel.leftChildren_apply]
rfl
def rightChild (record : Record (C := C) (t := t) rows clauses designated)
(i : Nonclean rows clauses designated record) :
SourceChildKernel.Block C rows (childSlots (rightRef rows clauses designated record) i.val) :=
cast (congrArg (SourceChildKernel.Block C rows) (rightRef_childSlots rows clauses designated record i).symm)
(physicalRight rows clauses designated i.val (extract rows clauses designated record i))
def leftChild (record : Record (C := C) (t := t) rows clauses designated)
(i : Nonclean rows clauses designated record) :
SourceChildKernel.Block C rows (childSlots (leftRef rows clauses designated record) i.val) :=
cast (congrArg (SourceChildKernel.Block C rows) (leftRef_childSlots rows clauses designated record i).symm)
(physicalLeft rows clauses designated i.val (extract rows clauses designated record i))
theorem rightChild_heq (record : Record (C := C) (t := t) rows clauses designated)
(i : Nonclean rows clauses designated record) :
HEq (rightChild rows clauses designated record i)
(physicalRight rows clauses designated i.val (extract rows clauses designated record i)) :=
cast_heq _ _
theorem leftChild_heq (record : Record (C := C) (t := t) rows clauses designated)
(i : Nonclean rows clauses designated record) :
HEq (leftChild rows clauses designated record i)
(physicalLeft rows clauses designated i.val (extract rows clauses designated record i)) :=
cast_heq _ _
private theorem dependent_congr {A : Sort*} {B : A → Sort*} (f : (a : A) → B a)
{a b : A} (h : a = b) : HEq (f a) (f b) := by
cases h
rfl
theorem rightChild_event_heq (record : Record (C := C) (t := t) rows clauses designated)
(x : Sample (C := C) (t := t) rows clauses designated)
(hevent : event rows clauses designated record x = true)
(i : Nonclean rows clauses designated record) :
HEq (rightChild rows clauses designated record i)
(SourceChildKernel.rightChildren rows clauses designated
(fun j => ((x j).1, (x j).2.1)) (fun j => (x j).2.2) i.val) :=
(rightChild_heq rows clauses designated record i).trans
(dependent_congr (physicalRight rows clauses designated i.val)
(extract_eq rows clauses designated record x hevent i))
theorem leftChild_event_heq (record : Record (C := C) (t := t) rows clauses designated)
(x : Sample (C := C) (t := t) rows clauses designated)
(hevent : event rows clauses designated record x = true)
(i : Nonclean rows clauses designated record) :
HEq (leftChild rows clauses designated record i)
(SourceChildKernel.leftChildren rows clauses designated
(fun j => ((x j).1, (x j).2.1)) (fun j => (x j).2.2) i.val) := by
have hphysical :
HEq (physicalLeft rows clauses designated i.val (extract rows clauses designated record i))
(physicalLeft rows clauses designated i.val (x i.val)) :=
dependent_congr (physicalLeft rows clauses designated i.val)
(extract_eq rows clauses designated record x hevent i)
exact (leftChild_heq rows clauses designated record i).trans
(hphysical.trans (heq_of_eq (physicalLeft_eq_leftChildren rows clauses designated x i.val)))
omit [NeZero m] in
theorem childClean_of_clean (record : Record (C := C) (t := t) rows clauses designated)
(x : Sample (C := C) (t := t) rows clauses designated)
(hevent : event rows clauses designated record x = true) (i : Fin (branch h))
(hi : clean rows clauses designated record i) :
UniformCleanSources.childClean
(SourceChildKernel.Factors (C := C) (t := t) rows clauses designated)
(SourceChildKernel.zero (C := C) (t := t) rows clauses designated)
i (x i).1 (x i).2.1 (x i).2.2 = true := by
have obs := CleanSourceRecord.observation_eq designated
(SourceChildKernel.Factors (C := C) (t := t) rows clauses designated)
(SourceChildKernel.zero (C := C) (t := t) rows clauses designated) record x hevent i
cases hc : UniformCleanSources.childClean
(SourceChildKernel.Factors (C := C) (t := t) rows clauses designated)
(SourceChildKernel.zero (C := C) (t := t) rows clauses designated)
i (x i).1 (x i).2.1 (x i).2.2 with
| false =>
have hr : record i = .inl (x i) := by
simpa only [SelectedSources.childObservation, hc, Bool.false_eq_true, ite_false]
using obs.symm
simp only [clean, CleanSourceRecord.clean, hr, UniformCleanSources.cleanRecord,
Bool.false_eq_true] at hi
| true => rfl
end
end PerfectCompleteness.CleanRecordChildBlocks
end
end OAI
|