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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 | import OAI.Computability.PerfectCompleteness.Decoding.FixedStoppedDecoderContext
import OAI.Computability.PerfectCompleteness.Sampling.CandidateCoupling
namespace OAI
section
namespace PerfectCompleteness.FixedStoppedDirections
noncomputable section
open scoped Classical
open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays
open UniqueGamesTheorem.Foundations.Games
open UniqueGamesTheorem.Appendix.RankLevelFilter (linearMapFintype)
attribute [local instance] linearMapFintype
variable {δ : ℚ} {hδ : 0 < δ} (parameters : FixedParameters.Parameters δ hδ)
(i j : Fin parameters.plan.depth) (hij : i < j)
abbrev lowerNode (pref : FixedStoppedDecoderContext.Prefixes parameters i j) :=
HierarchicalLeftDecoder.LowerNode (FixedStoppedDecoderContext.upper parameters i j pref)
(i.val + 1) (FixedStoppedDecoderContext.lower parameters i j hij pref)
@[simp] theorem lowerNode_height (pref : FixedStoppedDecoderContext.Prefixes parameters i j) :
Nodes.height (lowerNode parameters i j hij pref) = i.val + 1 :=
CleanDecoderContext.lowerHeight (FixedStoppedDecoderContext.upper parameters i j pref)
(FixedStoppedDecoderContext.lower parameters i j hij pref)
abbrev LowerDirection (pref : FixedStoppedDecoderContext.Prefixes parameters i j) :=
BucketSampler.Direction
(FixedRows.rows parameters.plan (Nodes.height (lowerNode parameters i j hij pref)))
instance lowerDirectionNonempty (pref : FixedStoppedDecoderContext.Prefixes parameters i j) :
Nonempty (LowerDirection parameters i j hij pref) :=
PreliminarySampler.directionNonempty _ (by
rw [lowerNode_height]
exact parameters.plan.rows_pos (parameters.plan.depth - (i.val + 1)))
def lowerDirectionEquiv (pref : FixedStoppedDecoderContext.Prefixes parameters i j) :
PrefixTests.LevelDirection (FixedRows.rows parameters.plan) i ≃
LowerDirection parameters i j hij pref :=
(Equiv.cast (congrArg (fun height =>
BucketSampler.Direction (FixedRows.rows parameters.plan height))
(lowerNode_height parameters i j hij pref))).symm
def lowerDirection (pref : FixedStoppedDecoderContext.Prefixes parameters i j)
(directions : FixedStoppedDecoderContext.Directions parameters) :
LowerDirection parameters i j hij pref :=
PrefixTests.nodeDirection (FixedRows.rows parameters.plan) i
⟨lowerNode parameters i j hij pref, lowerNode_height parameters i j hij pref⟩ (directions i)
theorem lowerDirection_eq_equiv (pref : FixedStoppedDecoderContext.Prefixes parameters i j)
(directions : FixedStoppedDecoderContext.Directions parameters) :
lowerDirection parameters i j hij pref directions =
lowerDirectionEquiv parameters i j hij pref (directions i) := rfl
def lowerLaw (pref : FixedStoppedDecoderContext.Prefixes parameters i j) :
FiniteDistribution (LowerDirection parameters i j hij pref) :=
FiniteDistribution.uniform _
theorem coordinate_law (level : Fin parameters.plan.depth) :
(FixedStoppedDecoderContext.directionLaw parameters).pushforward
(fun directions => directions level) =
CanonicalDirections.levelLaw (FixedRows.rows parameters.plan)
(fun k => parameters.plan.rows_pos (parameters.plan.depth - (k + 1))) level :=
FiniteProduct.eval_pushforward
(CanonicalDirections.levelLaw (FixedRows.rows parameters.plan)
(fun k => parameters.plan.rows_pos (parameters.plan.depth - (k + 1)))) level
theorem lowerDirection_law (pref : FixedStoppedDecoderContext.Prefixes parameters i j) :
(FixedStoppedDecoderContext.directionLaw parameters).pushforward
(lowerDirection parameters i j hij pref) = lowerLaw parameters i j hij pref := by
let : Nonempty (PrefixTests.LevelDirection (FixedRows.rows parameters.plan) i) :=
PreliminarySampler.directionNonempty _
(parameters.plan.rows_pos (parameters.plan.depth - (i.val + 1)))
change (FixedStoppedDecoderContext.directionLaw parameters).pushforward
(fun directions => lowerDirectionEquiv parameters i j hij pref (directions i)) = _
rw [← FiniteDistribution.pushforward_comp (FixedStoppedDecoderContext.directionLaw parameters)
(fun directions => directions i) (lowerDirectionEquiv parameters i j hij pref), coordinate_law]
change (FiniteDistribution.uniform (PrefixTests.LevelDirection (FixedRows.rows parameters.plan) i)).pushforward
(lowerDirectionEquiv parameters i j hij pref) =
FiniteDistribution.uniform (LowerDirection parameters i j hij pref)
rw [← FiniteDistribution.transport_eq_pushforward]
exact UniformConditioning.uniform_transport (lowerDirectionEquiv parameters i j hij pref)
theorem lowerDirection_expectation (pref : FixedStoppedDecoderContext.Prefixes parameters i j)
(f : LowerDirection parameters i j hij pref → ℝ) :
(FixedStoppedDecoderContext.directionLaw parameters).expectation
(fun directions => f (lowerDirection parameters i j hij pref directions)) =
(lowerLaw parameters i j hij pref).expectation f := by
rw [← lowerDirection_law, FiniteDistribution.expectation_pushforward]
variable (input : List Bool)
theorem direction_eq (context : FixedStoppedDecoderContext.Context parameters i j hij input) :
FixedStoppedDecoderContext.direction parameters i j hij input context =
(lowerDirection parameters i j hij context.1 context.2.2.2.1).val := rfl
theorem expectation_coordinate
(f : (pref : FixedStoppedDecoderContext.Prefixes parameters i j) →
(external : FixedStoppedDecoderContext.OutsideQuestions parameters i j hij input pref) →
FixedStoppedDecoderContext.Advice parameters i j pref →
FixedStoppedDecoderContext.Exterior parameters i j hij input pref external →
PrefixTests.LevelDirection (FixedRows.rows parameters.plan) i → ℝ) :
(FixedStoppedDecoderContext.law parameters i j hij input).expectation (fun context =>
f context.1 context.2.1 context.2.2.1 context.2.2.2.2 (context.2.2.2.1 i)) =
(FixedStoppedDecoderContext.prefixLaw parameters i j hij).expectation (fun pref =>
(FiniteDistribution.uniform
(FixedStoppedDecoderContext.OutsideQuestions parameters i j hij input pref)).expectation
(fun external =>
(FiniteDistribution.uniform (FixedStoppedDecoderContext.Advice parameters i j pref)).expectation
(fun A =>
(FiniteDistribution.uniform
(FixedStoppedDecoderContext.Exterior parameters i j hij input pref external)).expectation
(fun exterior =>
(CanonicalDirections.levelLaw (FixedRows.rows parameters.plan)
(fun k => parameters.plan.rows_pos (parameters.plan.depth - (k + 1))) i).expectation
(fun direction => f pref external A exterior direction))))) := by
rw [FixedStoppedDecoderContext.law, CandidateCoupling.expectation_sigmaLaw]
apply FiniteDistribution.expectation_congr
intro pref
rw [CandidateCoupling.expectation_sigmaLaw]
apply FiniteDistribution.expectation_congr
intro external
rw [FixedStoppedDecoderContext.fiberLaw, FiniteDistribution.expectation_product]
apply FiniteDistribution.expectation_congr
intro A
rw [FiniteDistribution.expectation_product, FiniteDistribution.expectation_comm]
apply FiniteDistribution.expectation_congr
intro exterior
rw [← coordinate_law parameters i, FiniteDistribution.expectation_pushforward]
theorem expectation_lowerDirection
(f : (pref : FixedStoppedDecoderContext.Prefixes parameters i j) →
(external : FixedStoppedDecoderContext.OutsideQuestions parameters i j hij input pref) →
FixedStoppedDecoderContext.Advice parameters i j pref →
FixedStoppedDecoderContext.Exterior parameters i j hij input pref external →
LowerDirection parameters i j hij pref → ℝ) :
(FixedStoppedDecoderContext.law parameters i j hij input).expectation (fun context =>
f context.1 context.2.1 context.2.2.1 context.2.2.2.2
(lowerDirection parameters i j hij context.1 context.2.2.2.1)) =
(FixedStoppedDecoderContext.prefixLaw parameters i j hij).expectation (fun pref =>
(FiniteDistribution.uniform
(FixedStoppedDecoderContext.OutsideQuestions parameters i j hij input pref)).expectation
(fun external =>
(FiniteDistribution.uniform (FixedStoppedDecoderContext.Advice parameters i j pref)).expectation
(fun A =>
(FiniteDistribution.uniform
(FixedStoppedDecoderContext.Exterior parameters i j hij input pref external)).expectation
(fun exterior =>
(lowerLaw parameters i j hij pref).expectation
(fun direction => f pref external A exterior direction))))) := by
rw [FixedStoppedDecoderContext.law, CandidateCoupling.expectation_sigmaLaw]
apply FiniteDistribution.expectation_congr
intro pref
rw [CandidateCoupling.expectation_sigmaLaw]
apply FiniteDistribution.expectation_congr
intro external
rw [FixedStoppedDecoderContext.fiberLaw, FiniteDistribution.expectation_product]
apply FiniteDistribution.expectation_congr
intro A
rw [FiniteDistribution.expectation_product, FiniteDistribution.expectation_comm]
apply FiniteDistribution.expectation_congr
intro exterior
exact lowerDirection_expectation parameters i j hij pref (f pref external A exterior)
theorem expectation_direction
(f : (pref : FixedStoppedDecoderContext.Prefixes parameters i j) →
(external : FixedStoppedDecoderContext.OutsideQuestions parameters i j hij input pref) →
FixedStoppedDecoderContext.Advice parameters i j pref →
FixedStoppedDecoderContext.Exterior parameters i j hij input pref external →
Block (FixedRows.rows parameters.plan) (lowerNode parameters i j hij pref) → ℝ) :
(FixedStoppedDecoderContext.law parameters i j hij input).expectation (fun context =>
f context.1 context.2.1 context.2.2.1 context.2.2.2.2
(FixedStoppedDecoderContext.direction parameters i j hij input context)) =
(FixedStoppedDecoderContext.prefixLaw parameters i j hij).expectation (fun pref =>
(FiniteDistribution.uniform
(FixedStoppedDecoderContext.OutsideQuestions parameters i j hij input pref)).expectation
(fun external =>
(FiniteDistribution.uniform (FixedStoppedDecoderContext.Advice parameters i j pref)).expectation
(fun A =>
(FiniteDistribution.uniform
(FixedStoppedDecoderContext.Exterior parameters i j hij input pref external)).expectation
(fun exterior =>
(lowerLaw parameters i j hij pref).expectation
(fun direction => f pref external A exterior direction.val))))) :=
expectation_lowerDirection parameters i j hij input
(fun pref external A exterior direction => f pref external A exterior direction.val)
end
end PerfectCompleteness.FixedStoppedDirections
end
end OAI
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