File size: 10,611 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 | import OAI.Computability.PerfectCompleteness.Decoding.CleanDecoderRateLemmas
import OAI.Computability.PerfectCompleteness.Repetition.CleanPhysicalCanonicalQuery
namespace OAI
section
namespace PerfectCompleteness.CleanNativeReplay
noncomputable section
open scoped Classical
open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays
open UniqueGamesTheorem.Foundations.Games
section Transport
variable {branch : Nat → Nat} {n h t : Nat}
private theorem transport_target_heq (rows repeats : Nat → Nat)
(p : Path branch n (h + 1))
(reference target other : Slots branch n → Fin t → MixedSupport.Slot)
(clean : Fin (branch h) → Prop) (heq : target = other)
(hs : ∀ s, CutSamplerKeyLocality.keptLeaf p clean s → reference s = target s)
(hs' : ∀ s, CutSamplerKeyLocality.keptLeaf p clean s → reference s = other s)
(tape : WholeCutReplay.Tape rows repeats p reference clean) :
HEq (WholeCutReplayKeys.transportTape rows repeats p reference target clean hs tape)
(WholeCutReplayKeys.transportTape rows repeats p reference other clean hs' tape) := by
cases heq
rfl
private theorem evaluateArrays_heq (rows repeats : Nat → Nat)
(p : Path branch n (h + 1))
(slots target : Slots branch n → Fin t → MixedSupport.Slot)
(clean : Fin (branch h) → Prop) (hs : slots = target)
(tape : WholeCutReplay.Tape rows repeats p slots clean)
(other : WholeCutReplay.Tape rows repeats p target clean) (htape : HEq tape other) :
HEq (WholeCutReplay.evaluateArrays rows repeats p slots clean tape)
(WholeCutReplay.evaluateArrays rows repeats p target clean other) := by
cases hs
cases htape
rfl
end Transport
variable {branch rows repeats : Nat → Nat} {n h t v m : Nat} [NeZero m]
{upper : Nodes branch n} {d : HierarchicalFrozenTables.LowerNodes upper (h + 1)}
(S : CleanDecoderRate.Setup branch rows repeats n h t v m upper d)
abbrev context (x : CleanDecoderRate.Sample S) := CleanDecoderPairLaw.context S x
abbrev leftOwn (x : CleanDecoderRate.Sample S) := CleanDecoderPairLaw.leftOwn S x
abbrev rightOwn (x : CleanDecoderRate.Sample S) := CleanDecoderPairLaw.rightOwn S x
abbrev leftSlots (x : CleanDecoderRate.Sample S) :=
CleanPhysicalCanonicalQuery.leftSlots rows repeats (CleanDecoderRate.path S)
S.outside S.clauses S.designated x
abbrev rightSlots (x : CleanDecoderRate.Sample S) :=
CleanPhysicalCanonicalQuery.rightSlots rows repeats (CleanDecoderRate.path S)
S.outside S.clauses S.designated x
abbrev leftTape (x : CleanDecoderRate.Sample S) :=
CleanPhysicalReplay.leftTape rows repeats (CleanDecoderRate.path S)
S.outside S.placeholder S.clauses S.designated x S.exterior
abbrev rightTape (x : CleanDecoderRate.Sample S) :=
CleanPhysicalReplay.rightTape rows repeats (CleanDecoderRate.path S)
S.outside S.placeholder S.clauses S.designated x S.exterior
theorem leftSlots_eq (x : CleanDecoderRate.Sample S) :
CleanDecoderContext.leftSlots (context S x) (leftOwn S x) = leftSlots S x :=
CleanRecordDecoderInput.leftSlots_eq rows repeats upper d S.cut S.outside S.placeholder
S.clauses S.designated (CleanDecoderRate.observations S x) S.exterior x
(CleanRecordDecoderInput.event_observe rows repeats upper d S.cut S.clauses S.designated x)
theorem rightSlots_eq (x : CleanDecoderRate.Sample S) :
CleanDecoderContext.rightSlots (context S x) (rightOwn S x) = rightSlots S x :=
CleanRecordDecoderInput.rightSlots_eq rows repeats upper d S.cut S.outside S.placeholder
S.clauses S.designated (CleanDecoderRate.observations S x) S.exterior x
(CleanRecordDecoderInput.event_observe rows repeats upper d S.cut S.clauses S.designated x)
theorem leftReplay_heq (x : CleanDecoderRate.Sample S) :
HEq (CleanLeftDecoder.replay (CleanDecoderContext.leftExposed (context S x)) (leftOwn S x))
(WholeCutReplay.erase rows repeats (CleanDecoderRate.path S) (leftSlots S x)
(context S x).clean (leftTape S x)) := by
let record := CleanDecoderRate.observations S x
have hevent := CleanRecordDecoderInput.event_observe
rows repeats upper d S.cut S.clauses S.designated x
have hrecord := CleanPhysicalReplay.leftRecord_eq_erase
rows repeats (CleanDecoderRate.path S) S.outside S.placeholder S.clauses S.designated
record x hevent S.exterior
have htransport := transport_target_heq rows repeats (CleanDecoderRate.path S)
(CutSlotAssembly.fill (CleanDecoderRate.path S) S.outside
(CleanRecordChildBlocks.leftRef rows S.clauses S.designated record))
(CleanDecoderContext.leftSlots (context S x) (leftOwn S x)) (leftSlots S x)
(context S x).clean (leftSlots_eq S x)
(CleanEndpointSlots.leftSlots_kept_eq S.clauses S.designated (context S x).clean
(context S x).visible (CleanDecoderRate.path S) S.outside (fun _ _ => 0) (leftOwn S x))
(CutSlotAssembly.fill_kept_eq (CleanDecoderRate.path S) S.outside _ _ _
(CleanPhysicalReplay.leftInside_nonclean rows repeats (CleanDecoderRate.path S)
S.clauses S.designated record x hevent))
(CleanPhysicalReplay.leftRecord rows repeats (CleanDecoderRate.path S)
S.outside S.placeholder S.clauses S.designated record S.exterior)
exact htransport.trans (heq_of_eq hrecord)
theorem rightReplay_heq (x : CleanDecoderRate.Sample S) :
HEq (CleanRightDecoder.replay (context S x).clauses (context S x).designated
(context S x).clean (context S x).visible (context S x).projected upper
(CleanDecoderContext.lower (context S x)) (context S x).cut
(CleanDecoderContext.lowerHeight upper d) (context S x).outside rows repeats
(context S x).rightTape (rightOwn S x))
(WholeCutReplay.erase rows repeats (CleanDecoderRate.path S) (rightSlots S x)
(context S x).clean (rightTape S x)) := by
let record := CleanDecoderRate.observations S x
have hevent := CleanRecordDecoderInput.event_observe
rows repeats upper d S.cut S.clauses S.designated x
have hrecord := CleanPhysicalReplay.rightRecord_eq_erase
rows repeats (CleanDecoderRate.path S) S.outside S.placeholder S.clauses S.designated
record x hevent S.exterior
have htransport := transport_target_heq rows repeats (CleanDecoderRate.path S)
(CutSlotAssembly.fill (CleanDecoderRate.path S) S.outside
(CleanRecordChildBlocks.rightRef rows S.clauses S.designated record))
(CleanDecoderContext.rightSlots (context S x) (rightOwn S x)) (rightSlots S x)
(context S x).clean (rightSlots_eq S x)
(CleanEndpointSlots.rightSlots_kept_eq S.clauses S.designated (context S x).clean
(context S x).visible (CleanDecoderRate.path S) S.outside (context S x).projected
(fun _ _ => (S.clauses 0).variable 0) (rightOwn S x))
(CutSlotAssembly.fill_kept_eq (CleanDecoderRate.path S) S.outside _ _ _
(CleanPhysicalReplay.rightInside_nonclean rows repeats (CleanDecoderRate.path S)
S.clauses S.designated record x hevent))
(CleanPhysicalReplay.rightRecord rows repeats (CleanDecoderRate.path S)
S.outside S.placeholder S.clauses S.designated record S.exterior)
exact htransport.trans (heq_of_eq hrecord)
variable (cube : Nat) (hcube : 0 < cube)
theorem left_grouped_zero (x : CleanDecoderRate.Sample S)
(hx : (CleanDecoderRate.rawLaw S cube hcube).weight x ≠ 0) :
WholeCutZero.GroupedAtClean rows repeats (CleanDecoderRate.path S) (leftSlots S x)
(context S x).clean (leftTape S x) :=
CleanPhysicalReplay.leftTape_grouped_zero rows repeats (CleanDecoderRate.path S)
S.outside S.placeholder S.clauses S.designated
(fun _ => SourceOddLists.tupleLaw m t)
(fun i => FiniteProduct.law
(fun _ : {leaf : Slots branch h // leaf ≠ S.designated i} => SourceOddLists.tupleLaw m t))
(fun _ => ProjectedCleanRate.projectionFlag cube hcube)
(CleanDecoderRate.observations S x) x hx
(CleanRecordDecoderInput.event_observe rows repeats upper d S.cut S.clauses S.designated x)
S.exterior
theorem right_grouped_zero (x : CleanDecoderRate.Sample S)
(hx : (CleanDecoderRate.rawLaw S cube hcube).weight x ≠ 0) :
WholeCutZero.GroupedAtClean rows repeats (CleanDecoderRate.path S) (rightSlots S x)
(context S x).clean (rightTape S x) :=
CleanPhysicalReplay.rightTape_grouped_zero rows repeats (CleanDecoderRate.path S)
S.outside S.placeholder S.clauses S.designated
(fun _ => SourceOddLists.tupleLaw m t)
(fun i => FiniteProduct.law
(fun _ : {leaf : Slots branch h // leaf ≠ S.designated i} => SourceOddLists.tupleLaw m t))
(fun _ => ProjectedCleanRate.projectionFlag cube hcube)
(CleanDecoderRate.observations S x) x hx
(CleanRecordDecoderInput.event_observe rows repeats upper d S.cut S.clauses S.designated x)
S.exterior
theorem leftArrays_heq (x : CleanDecoderRate.Sample S)
(hx : (CleanDecoderRate.rawLaw S cube hcube).weight x ≠ 0) :
HEq (CleanLeftDecoder.arrays (CleanDecoderContext.leftExposed (context S x)) (leftOwn S x))
(WholeCutSampler.evaluate rows repeats (CleanDecoderRate.path S) (leftSlots S x)
(leftTape S x)) := by
have h := evaluateArrays_heq rows repeats (CleanDecoderRate.path S)
(CleanDecoderContext.leftSlots (context S x) (leftOwn S x)) (leftSlots S x)
(context S x).clean (leftSlots_eq S x) _ _ (leftReplay_heq S x)
exact h.trans (heq_of_eq (WholeCutReplay.evaluateArrays_erase_of_grouped_zero
rows repeats (CleanDecoderRate.path S) (leftSlots S x) (context S x).clean
(leftTape S x) (left_grouped_zero S cube hcube x hx)))
theorem rightArrays_heq (x : CleanDecoderRate.Sample S)
(hx : (CleanDecoderRate.rawLaw S cube hcube).weight x ≠ 0) :
HEq (CleanRightDecoder.arrays (context S x).clauses (context S x).designated
(context S x).clean (context S x).visible (context S x).projected upper
(CleanDecoderContext.lower (context S x)) (context S x).cut
(CleanDecoderContext.lowerHeight upper d) (context S x).outside rows repeats
(context S x).rightTape (rightOwn S x))
(WholeCutSampler.evaluate rows repeats (CleanDecoderRate.path S) (rightSlots S x)
(rightTape S x)) := by
have h := evaluateArrays_heq rows repeats (CleanDecoderRate.path S)
(CleanDecoderContext.rightSlots (context S x) (rightOwn S x)) (rightSlots S x)
(context S x).clean (rightSlots_eq S x) _ _ (rightReplay_heq S x)
exact h.trans (heq_of_eq (WholeCutReplay.evaluateArrays_erase_of_grouped_zero
rows repeats (CleanDecoderRate.path S) (rightSlots S x) (context S x).clean
(rightTape S x) (right_grouped_zero S cube hcube x hx)))
end
end PerfectCompleteness.CleanNativeReplay
end
end OAI
|