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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 | import OAI.Computability.PerfectCompleteness.Construction.SourceChildKernelLemmas
import OAI.Computability.PerfectCompleteness.Foundations.DependentPredictionDifferenceLemmas
import OAI.Computability.PerfectCompleteness.Sampling.SourceChildKernelMarginalLemmas
import OAI.Computability.PerfectCompleteness.Sampling.SourceChildQuestionLaw
namespace OAI
section
namespace PerfectCompleteness.SourcePhysicalAssemblyLaw
open UniqueGamesTheorem.Foundations.Games
open RecursiveSpaces TreeSourceSpaces SourceChildKernel
open scoped Classical
noncomputable section
private theorem kernelJoint_pushforward_sigma {S R : Type*} {Y : S → Type*}
[Fintype S] [Fintype R] [∀ s, Fintype (Y s)]
(μ : FiniteDistribution S) (P : S → FiniteDistribution R)
(f : (s : S) → R → Y s) :
(CleanConditioning.kernelJoint μ P).pushforward
(fun x => (⟨x.1, f x.1 x.2⟩ : Σ s, Y s)) =
CompletionSoundness.sigmaLaw μ (fun s => (P s).pushforward (f s)) := by
apply SigmaObservation.eq_of_probability_eq
intro event
rw [FiniteDistribution.probability_pushforward, CleanConditioning.probability_kernelJoint,
CompletionSoundness.sigmaLaw_probability]
simp only [FiniteDistribution.expectation, FiniteDistribution.probability_pushforward]
variable {branch : Nat → Nat} {n t v m : Nat} {C : Type*} [Fintype C]
(rows : Nat → Nat) (clauses : Fin m → SourceClause.NormalizedClause v)
(designated : Fin (branch n) → Slots branch n)
def assembleLeft (sources : Sources (m := m) (t := t) designated)
(raw : (i : Fin (branch n)) → Raw (C := C) (t := t) rows clauses designated i) :
CutChildGrouping.Assembled (C := C) (parentLeftSlots clauses designated sources) rows :=
CutChildGrouping.assemble (parentLeftSlots clauses designated sources) rows
(leftChildren rows clauses designated sources raw)
theorem assembleLeft_eq_sourcePullback
(sources : Sources (m := m) (t := t) designated)
(raw : (i : Fin (branch n)) → Raw (C := C) (t := t) rows clauses designated i) :
assembleLeft rows clauses designated sources raw =
ChildAssemblyProjection.assembledPullback rows
(parentProjection rows clauses designated sources raw)
(CutChildGrouping.assemble (parentRightSlots rows clauses designated sources raw) rows
(rightChildren rows clauses designated sources raw)) :=
SourceChildKernel.assemble_leftChildren rows clauses designated sources raw
theorem kernels_assembleLeft (flag : Fin (branch n) → FiniteDistribution Bool)
(sources : Sources (m := m) (t := t) designated) :
(FiniteProduct.law (fun i =>
kernel (C := C) (t := t) rows clauses designated flag i (sources i))).pushforward
(assembleLeft rows clauses designated sources) =
(FiniteProduct.law (fun i =>
SourceChildNativeLaw.leftLaw (C := C) (t := t) rows clauses designated flag i (sources i))).pushforward
(CutChildGrouping.assemble (C := C) (parentLeftSlots clauses designated sources) rows) := by
unfold assembleLeft
rw [← FiniteDistribution.pushforward_comp
(FiniteProduct.law (fun i =>
kernel (C := C) (t := t) rows clauses designated flag i (sources i)))
(leftChildren rows clauses designated sources)
(CutChildGrouping.assemble (C := C) (parentLeftSlots clauses designated sources) rows),
SourceChildNativeLaw.kernels_leftChildren]
rfl
theorem kernels_assembleLeft_false
(sources : Sources (m := m) (t := t) designated) :
(FiniteProduct.law (fun i =>
kernel (C := C) (t := t) rows clauses designated
(fun _ => SourceChildNativeLaw.falseFlag) i (sources i))).pushforward
(assembleLeft rows clauses designated sources) =
FiniteDistribution.uniform
(CutChildGrouping.Assembled (C := C) (parentLeftSlots clauses designated sources) rows) := by
unfold assembleLeft
rw [← FiniteDistribution.pushforward_comp
(FiniteProduct.law (fun i =>
kernel (C := C) (t := t) rows clauses designated
(fun _ => SourceChildNativeLaw.falseFlag) i (sources i)))
(leftChildren rows clauses designated sources)
(CutChildGrouping.assemble (C := C) (parentLeftSlots clauses designated sources) rows),
SourceChildNativeLaw.kernels_leftChildren_false]
exact CutChildGrouping.assemble_law (C := C) (parentLeftSlots clauses designated sources) rows
abbrev Observation :=
Σ sources : Sources (m := m) (t := t) designated,
CutChildGrouping.Assembled (C := C) (parentLeftSlots clauses designated sources) rows
def observe (sample : Sample (C := C) (t := t) rows clauses designated) :
Observation (C := C) (t := t) rows clauses designated :=
⟨(SourceChildKernelJoint.groupEquiv rows clauses designated sample).1,
assembleLeft rows clauses designated
(SourceChildKernelJoint.groupEquiv rows clauses designated sample).1
(SourceChildKernelJoint.groupEquiv rows clauses designated sample).2⟩
private theorem original_observe_kernel_law [NeZero m]
(flag : Fin (branch n) → FiniteDistribution Bool) :
(originalLaw (C := C) (t := t) rows clauses designated flag).pushforward
(observe rows clauses designated) =
CompletionSoundness.sigmaLaw (SourceChildQuestionLaw.sourceLaw m t designated)
(fun sources =>
(FiniteProduct.law (fun i =>
kernel (C := C) (t := t) rows clauses designated flag i (sources i))).pushforward
(assembleLeft rows clauses designated sources)) := by
unfold observe
rw [← FiniteDistribution.pushforward_comp
(originalLaw (C := C) (t := t) rows clauses designated flag)
(SourceChildKernelJoint.groupEquiv rows clauses designated)
(fun x => (⟨x.1, assembleLeft rows clauses designated x.1 x.2⟩ :
Observation (C := C) (t := t) rows clauses designated)),
SourceChildKernelJoint.originalLaw_group]
exact kernelJoint_pushforward_sigma (SourceChildQuestionLaw.sourceLaw m t designated)
(fun sources => FiniteProduct.law (fun i =>
kernel (C := C) (t := t) rows clauses designated flag i (sources i)))
(assembleLeft rows clauses designated)
theorem original_assembleLeft_law [NeZero m]
(flag : Fin (branch n) → FiniteDistribution Bool) :
(originalLaw (C := C) (t := t) rows clauses designated flag).pushforward
(observe rows clauses designated) =
CompletionSoundness.sigmaLaw (SourceChildQuestionLaw.sourceLaw m t designated)
(fun sources =>
(FiniteProduct.law (fun i => SourceChildNativeLaw.leftLaw (C := C) (t := t)
rows clauses designated flag i (sources i))).pushforward
(CutChildGrouping.assemble (C := C) (parentLeftSlots clauses designated sources) rows)) := by
rw [original_observe_kernel_law]
exact congrArg (CompletionSoundness.sigmaLaw (SourceChildQuestionLaw.sourceLaw m t designated))
(funext fun sources => kernels_assembleLeft rows clauses designated flag sources)
theorem original_assembleLeft_false_law [NeZero m] :
(originalLaw (C := C) (t := t) rows clauses designated
(fun _ => SourceChildNativeLaw.falseFlag)).pushforward
(observe rows clauses designated) =
CompletionSoundness.sigmaLaw (SourceChildQuestionLaw.sourceLaw m t designated)
(fun sources => FiniteDistribution.uniform
(CutChildGrouping.Assembled (C := C) (parentLeftSlots clauses designated sources) rows)) := by
rw [original_observe_kernel_law]
exact congrArg (CompletionSoundness.sigmaLaw (SourceChildQuestionLaw.sourceLaw m t designated))
(funext fun sources => kernels_assembleLeft_false (C := C) rows clauses designated sources)
end
end PerfectCompleteness.SourcePhysicalAssemblyLaw
end
end OAI
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