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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 | import OAI.Computability.PerfectCompleteness.Decoding.LowerProjectedOriginalSuccessLemmas
import OAI.Computability.PerfectCompleteness.Repetition.CleanPhysicalReplayLaw
namespace OAI
section
namespace PerfectCompleteness.CleanPhysicalRawChildren
noncomputable section
open scoped Classical
open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays
open UniqueGamesTheorem.Foundations.Games
variable {branch : Nat → Nat} {n m t : Nat}
(rows repeats : Nat → Nat) (p : Path branch n (m + 1))
(outside : Slots branch n → Fin t → MixedSupport.Slot)
(placeholder inside : Slots branch (m + 1) → Fin t → MixedSupport.Slot)
def fillChildren
(children : CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p) inside rows) :
CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p)
(WholeCutGrouping.cutSlots p (CutSlotAssembly.fill p outside inside)) rows :=
fun i => CleanPhysicalReplay.fillChild rows repeats p outside inside i (children i)
theorem fillChildren_eq_childrenEquiv :
fillChildren rows repeats p outside inside =
(FilledUniformSampler.childrenEquiv rows repeats p outside inside : _ → _) := by
funext children i
exact eq_of_heq ((CleanPhysicalReplay.fillChild_heq rows repeats p outside inside i
(children i)).trans
(FilledUniformSampler.childrenEquiv_heq rows repeats p outside inside children i).symm)
theorem fillChildren_law :
(CutChildGrouping.rawLaw (C := WholeCutCalls.Index rows repeats p) inside rows).pushforward
(fillChildren rows repeats p outside inside) =
CutChildGrouping.rawLaw (C := WholeCutCalls.Index rows repeats p)
(WholeCutGrouping.cutSlots p (CutSlotAssembly.fill p outside inside)) rows := by
rw [fillChildren_eq_childrenEquiv]
simp only [CutChildGrouping.rawLaw, UniformLinearImage.law_uniform]
rw [FiniteDistribution.pushforward_equiv]
apply FiniteDistribution.eq_of_weight_eq
intro children
change 1 / (Fintype.card
(CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p) inside rows) : ℝ) =
1 / (Fintype.card (CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p)
(WholeCutGrouping.cutSlots p (CutSlotAssembly.fill p outside inside)) rows) : ℝ)
rw [Fintype.card_congr (FilledUniformSampler.childrenEquiv rows repeats p outside inside)]
theorem expectation_physicalTape
(external : CleanPhysicalReplay.Exterior rows repeats p outside placeholder)
(statistic : WholeCutSampler.Tape rows repeats p (CutSlotAssembly.fill p outside inside) → ℝ) :
(CutChildGrouping.rawLaw (C := WholeCutCalls.Index rows repeats p) inside rows).expectation
(fun children => statistic
(CleanPhysicalReplay.physicalTape rows repeats p outside placeholder inside external children)) =
(CutChildGrouping.rawLaw (C := WholeCutCalls.Index rows repeats p)
(WholeCutGrouping.cutSlots p (CutSlotAssembly.fill p outside inside)) rows).expectation
(fun children => statistic
((WholeCutGrouping.splitTape rows repeats p (CutSlotAssembly.fill p outside inside)).symm
(CleanPhysicalReplay.exteriorAt rows repeats p outside placeholder inside external,
children))) := by
have h := congrArg
(fun law : FiniteDistribution (CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p)
(WholeCutGrouping.cutSlots p (CutSlotAssembly.fill p outside inside)) rows) =>
law.expectation (fun children => statistic
((WholeCutGrouping.splitTape rows repeats p (CutSlotAssembly.fill p outside inside)).symm
(CleanPhysicalReplay.exteriorAt rows repeats p outside placeholder inside external,
children))))
(fillChildren_law rows repeats p outside inside)
rw [FiniteDistribution.expectation_pushforward] at h
exact h
theorem expectation_lowerTape (upper : Nodes branch n)
(pDown : Path branch (Nodes.height upper) (m + 1))
(external : CleanPhysicalReplay.Exterior rows repeats
((Nodes.path upper).append pDown) outside placeholder)
(statistic : WholeCutSampler.Tape rows repeats ((Nodes.path upper).append pDown)
(CutSlotAssembly.fill ((Nodes.path upper).append pDown) outside inside) → ℝ) :
(CutChildGrouping.rawLaw (C := WholeCutCalls.Index rows repeats ((Nodes.path upper).append pDown))
inside rows).expectation (fun children => statistic
(CleanPhysicalReplay.physicalTape rows repeats ((Nodes.path upper).append pDown)
outside placeholder inside external children)) =
(CutChildGrouping.rawLaw
(C := LowerCutPair.Calls rows repeats ((Nodes.path upper).append pDown))
(WholeCutGrouping.cutSlots ((Nodes.path upper).append pDown)
(CutSlotAssembly.fill ((Nodes.path upper).append pDown) outside inside)) rows).expectation
(fun children => statistic
(LowerPhysicalFiberSuccess.tape
(CutSlotAssembly.fill ((Nodes.path upper).append pDown) outside inside) upper pDown
(CleanPhysicalReplay.exteriorAt rows repeats ((Nodes.path upper).append pDown)
outside placeholder inside external) children)) :=
expectation_physicalTape rows repeats ((Nodes.path upper).append pDown)
outside placeholder inside external statistic
end
end PerfectCompleteness.CleanPhysicalRawChildren
end
end OAI
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