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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 | import OAI.Computability.PerfectCompleteness.Repetition.CleanPhysicalReplayLaw
import OAI.Computability.PerfectCompleteness.Sampling.RationalFiniteLawLemmas
import OAI.Computability.PerfectCompleteness.Sampling.SourceChildKernelMarginalLemmas
namespace OAI
section
namespace PerfectCompleteness.SourcePhysicalWholeLaw
open scoped Classical
open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays
open UniqueGamesTheorem.Foundations.Games
noncomputable section
variable {branch : Nat → Nat} {root h t v m : Nat}
(rows repeats : Nat → Nat) (p : Path branch root (h + 1))
(outside : Slots branch root → Fin t → MixedSupport.Slot)
(placeholder : Slots branch (h + 1) → Fin t → MixedSupport.Slot)
(clauses : Fin m → SourceClause.NormalizedClause v)
(designated : Fin (branch h) → Slots branch h)
abbrev Raw := CleanPhysicalReplay.Exterior rows repeats p outside placeholder ×
((i : Fin (branch h)) → SourceChildKernel.Raw
(C := WholeCutCalls.Index rows repeats p) (t := t) rows clauses designated i)
def rawLaw (flag : Fin (branch h) → FiniteDistribution Bool)
(sources : SourceChildKernel.Sources (m := m) (t := t) designated) :
FiniteDistribution (Raw rows repeats p outside placeholder clauses designated) :=
(FiniteDistribution.uniform (CleanPhysicalReplay.Exterior rows repeats p outside placeholder)).product
(FiniteProduct.law (fun i => SourceChildKernel.kernel
(C := WholeCutCalls.Index rows repeats p) (t := t) rows clauses designated flag i (sources i)))
def nativeBlockLaw (flag : Fin (branch h) → FiniteDistribution Bool)
(sources : SourceChildKernel.Sources (m := m) (t := t) designated) :
FiniteDistribution (CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p)
(SourceChildKernel.parentLeftSlots clauses designated sources) rows) :=
FiniteProduct.law (fun i => SourceChildNativeLaw.leftLaw
(C := WholeCutCalls.Index rows repeats p) (t := t) rows clauses designated flag i (sources i))
def observeLeft (sources : SourceChildKernel.Sources (m := m) (t := t) designated)
(sample : Raw rows repeats p outside placeholder clauses designated) :
Arrays (CutSlotAssembly.fill p outside (SourceChildKernel.parentLeftSlots clauses designated sources)) rows :=
WholeCutSampler.evaluate rows repeats p
(CutSlotAssembly.fill p outside (SourceChildKernel.parentLeftSlots clauses designated sources))
(CleanPhysicalReplay.physicalTape rows repeats p outside placeholder
(SourceChildKernel.parentLeftSlots clauses designated sources) sample.1
(SourceChildKernel.leftChildren rows clauses designated sources sample.2))
theorem observeLeft_law (flag : Fin (branch h) → FiniteDistribution Bool)
(sources : SourceChildKernel.Sources (m := m) (t := t) designated) :
(rawLaw rows repeats p outside placeholder clauses designated flag sources).pushforward
(observeLeft rows repeats p outside placeholder clauses designated sources) =
((FiniteDistribution.uniform (CleanPhysicalReplay.Exterior rows repeats p outside placeholder)).product
(nativeBlockLaw rows repeats p clauses designated flag sources)).pushforward
(fun x => WholeCutSampler.evaluate rows repeats p
(CutSlotAssembly.fill p outside (SourceChildKernel.parentLeftSlots clauses designated sources))
(CleanPhysicalReplay.physicalTape rows repeats p outside placeholder
(SourceChildKernel.parentLeftSlots clauses designated sources) x.1 x.2)) := by
let decode := fun sample : Raw rows repeats p outside placeholder clauses designated =>
(sample.1, SourceChildKernel.leftChildren rows clauses designated sources sample.2)
let evaluate := fun x : CleanPhysicalReplay.Exterior rows repeats p outside placeholder ×
CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p)
(SourceChildKernel.parentLeftSlots clauses designated sources) rows =>
WholeCutSampler.evaluate rows repeats p
(CutSlotAssembly.fill p outside (SourceChildKernel.parentLeftSlots clauses designated sources))
(CleanPhysicalReplay.physicalTape rows repeats p outside placeholder
(SourceChildKernel.parentLeftSlots clauses designated sources) x.1 x.2)
calc
_ = ((rawLaw rows repeats p outside placeholder clauses designated flag sources).pushforward
decode).pushforward evaluate := (FiniteDistribution.pushforward_comp _ _ _).symm
_ = _ := by
have hdecode : (rawLaw rows repeats p outside placeholder clauses designated flag sources).pushforward
decode =
(FiniteDistribution.uniform (CleanPhysicalReplay.Exterior rows repeats p outside placeholder)).product
(nativeBlockLaw rows repeats p clauses designated flag sources) := by
unfold rawLaw
change ((FiniteDistribution.uniform
(CleanPhysicalReplay.Exterior rows repeats p outside placeholder)).product
(FiniteProduct.law (fun i => SourceChildKernel.kernel
(C := WholeCutCalls.Index rows repeats p) (t := t) rows clauses designated flag i (sources i)))).pushforward
(fun sample => ((id : CleanPhysicalReplay.Exterior rows repeats p outside placeholder →
CleanPhysicalReplay.Exterior rows repeats p outside placeholder) sample.1,
SourceChildKernel.leftChildren rows clauses designated sources sample.2)) = _
rw [FiniteDistribution.product_pushforward,
UniqueGamesTheorem.Foundations.Games.FiniteDistribution.pushforward_id
(FiniteDistribution.uniform (CleanPhysicalReplay.Exterior rows repeats p outside placeholder)),
SourceChildNativeLaw.kernels_leftChildren]
rfl
rw [hdecode]
theorem nativeBlockLaw_false
(sources : SourceChildKernel.Sources (m := m) (t := t) designated) :
nativeBlockLaw rows repeats p clauses designated (fun _ => SourceChildNativeLaw.falseFlag) sources =
CutChildGrouping.rawLaw (C := WholeCutCalls.Index rows repeats p)
(SourceChildKernel.parentLeftSlots clauses designated sources) rows := by
unfold nativeBlockLaw CutChildGrouping.rawLaw
apply congrArg FiniteProduct.law
funext i
exact SourceChildNativeLaw.leftLaw_false rows clauses designated i (sources i)
theorem observeLeft_false_law
(sources : SourceChildKernel.Sources (m := m) (t := t) designated) :
(rawLaw rows repeats p outside placeholder clauses designated
(fun _ => SourceChildNativeLaw.falseFlag) sources).pushforward
(observeLeft rows repeats p outside placeholder clauses designated sources) =
WholeArraySampler.law rows repeats p
(CutSlotAssembly.fill p outside (SourceChildKernel.parentLeftSlots clauses designated sources)) := by
rw [observeLeft_law, nativeBlockLaw_false]
exact CleanPhysicalReplayLaw.evaluate_physicalTape_law rows repeats p outside placeholder
(SourceChildKernel.parentLeftSlots clauses designated sources)
end
end PerfectCompleteness.SourcePhysicalWholeLaw
end
end OAI
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