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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 | import OAI.Computability.PerfectCompleteness.Foundations.OriginalWholeCutBridge
import OAI.Computability.PerfectCompleteness.Reduction.CompletionSoundness
import OAI.Computability.PerfectCompleteness.Sampling.OriginalUniformCut
namespace OAI
section
namespace PerfectCompleteness.NumberedUniformCut
open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays
open OriginalWholeCutTape OriginalWholeCut
open UniqueGamesTheorem.Foundations.Games
open scoped Classical
noncomputable section
attribute [local instance 2000] OriginalWholeCutLaw.valuesBelowFintype
private theorem sigma_product_pushforward {E A Γ : Type*}
[Fintype E] [Fintype A] [Fintype Γ]
(μ : FiniteDistribution E) (ν : FiniteDistribution A) (f : E × A → Γ) :
(CompletionSoundness.sigmaLaw μ (fun _ => ν)).pushforward
(fun z => f (z.1, z.2)) =
(μ.product ν).pushforward f := by
apply FiniteDistribution.eq_of_weight_eq
intro z
simp only [FiniteDistribution.pushforward, CompletionSoundness.sigmaLaw,
FiniteDistribution.product, Fintype.sum_sigma, Fintype.sum_prod_type]
variable {branch : Nat → Nat} {n m t : Nat}
def numberRecordEquiv (rows repeats : Nat → Nat) (p : Path branch n (m + 1))
(slots : Slots branch n → Fin t → MixedSupport.Slot) :
Record rows repeats p slots ≃ OriginalWholeCutBridge.NumberedRecord rows repeats p slots :=
Equiv.prodCongr (Equiv.refl (Exterior rows repeats p slots))
((Equiv.prodCongr
(Equiv.arrowCongr (callNumbering rows repeats p) (Equiv.refl (H (cutSlots p slots))))
(Equiv.refl (BelowArrays rows p slots))).trans
(OriginalWholeCutBridge.assembledEquiv
(OriginalCutCalls.count rows repeats n (m + 1)) rows (cutSlots p slots)))
@[simp] theorem numberRecordEquiv_apply (rows repeats : Nat → Nat)
(p : Path branch n (m + 1)) (slots : Slots branch n → Fin t → MixedSupport.Slot)
(record : Record rows repeats p slots) :
numberRecordEquiv rows repeats p slots record =
OriginalWholeCutBridge.numberRecord rows repeats p slots record := rfl
theorem reference_numberRecord_law (rows repeats : Nat → Nat)
(p : Path branch n (m + 1)) (slots : Slots branch n → Fin t → MixedSupport.Slot) :
(OriginalUniformCut.referenceLaw rows repeats p slots).pushforward
(OriginalWholeCutBridge.numberRecord rows repeats p slots) =
(exteriorLaw rows repeats p slots).product
(FiniteDistribution.uniform (CutChildGrouping.Assembled
(C := Fin (OriginalCutCalls.count rows repeats n (m + 1))) (cutSlots p slots) rows)) := by
rw [OriginalUniformCut.referenceLaw_uniform]
change (FiniteDistribution.uniform (Record rows repeats p slots)).pushforward
(numberRecordEquiv rows repeats p slots) = _
rw [← FiniteDistribution.transport_eq_pushforward, UniformConditioning.uniform_transport]
exact WholeCutSampler.uniform_product.symm
theorem reconstruct_product_uniform (rows repeats : Nat → Nat)
(p : Path branch n (m + 1)) (slots : Slots branch n → Fin t → MixedSupport.Slot) :
((exteriorLaw rows repeats p slots).product
(FiniteDistribution.uniform (CutChildGrouping.Assembled
(C := Fin (OriginalCutCalls.count rows repeats n (m + 1))) (cutSlots p slots) rows))).pushforward
(OriginalWholeCutBridge.reconstruct rows repeats p slots) =
WholeArraySampler.law rows repeats p slots := by
rw [← reference_numberRecord_law rows repeats p slots, FiniteDistribution.pushforward_comp]
have hmap : (fun record : Record rows repeats p slots =>
OriginalWholeCutBridge.reconstruct rows repeats p slots
(OriginalWholeCutBridge.numberRecord rows repeats p slots record)) =
reconstructRecord rows repeats p slots :=
funext (OriginalWholeCutBridge.reconstruct_numberRecord rows repeats p slots)
rw [hmap]
exact OriginalUniformCut.reconstruct_referenceLaw rows repeats p slots
theorem reconstruct_sigma_uniform (rows repeats : Nat → Nat)
(p : Path branch n (m + 1)) (slots : Slots branch n → Fin t → MixedSupport.Slot) :
(CompletionSoundness.sigmaLaw (exteriorLaw rows repeats p slots) (fun _ =>
FiniteDistribution.uniform (CutChildGrouping.Assembled
(C := Fin (OriginalCutCalls.count rows repeats n (m + 1))) (cutSlots p slots) rows))).pushforward
(fun z => OriginalWholeCutBridge.reconstruct rows repeats p slots (z.1, z.2)) =
WholeArraySampler.law rows repeats p slots := by
rw [sigma_product_pushforward]
exact reconstruct_product_uniform rows repeats p slots
end
end PerfectCompleteness.NumberedUniformCut
end
end OAI
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