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import OAI.Computability.UniqueGames.Decoding.VisiblePoliciesLemmas
namespace OAI
/-!
# Actual conditional selected-fiber agreement
The selected event is the full-table event used by the total-variation
comparison. On a positive complete-data fiber it becomes the uniform actual
affine row/column slice. The selector is fixed by the left observation; the
extra complete data and a supporting matrix occur only in this analysis.
-/
noncomputable section
namespace UniqueGamesTheorem.Decoder.SelectedFiberLaw
open UniqueGamesTheorem.Integration.BinaryLinear UniqueGamesTheorem.Reduction UniqueGamesTheorem.Soundness
open UniqueGamesTheorem.Inverse UniqueGamesTheorem.Foundations.Games
open ActualSource VisiblePolicies SelectedPolicies MatrixCoordinates ActualSeedEvents
open scoped BigOperators Classical
attribute [local instance] Classical.propDecidable
attribute [local instance] Fintype.ofFinite
local instance homFintype {D F : Type*}
[AddCommGroup D] [Module F2 D] [AddCommGroup F] [Module F2 F]
[Fintype D] [Fintype F] : Fintype (D →ₗ[F2] F) :=
Fintype.ofInjective (fun M : D →ₗ[F2] F => (M : D → F)) DFunLike.coe_injective
/-- The universe finset is independent of the chosen finite enumeration.
The instances are explicit so elaboration need not unfold either enumeration. -/
private theorem univ_congr {X : Type*} (i j : Fintype X) :
@Finset.univ X i = @Finset.univ X j := by
cases Subsingleton.elim i j
rfl
variable {k s d r : ℕ}
def datumLeft (S : Source) (data : Datum S k s d r) : LeftInput S k s d (Vector r) :=
AdviceExperiment.leftObservation (ActualGame.rhs S) (AdviceLaw.completeDraw data 0)
def datumMatrix (S : Source) (data : Datum S k s d r) (N : AdviceLaw.PrivateMap data) :=
(AdviceLaw.completeDraw data N).hiddenMatrix
def datumProjection (S : Source) (data : Datum S k s d r) :=
AdviceExperiment.projection (ActualGame.rhs S) (AdviceLaw.completeDraw data 0)
theorem datumLeft_eq (S : Source) (data : Datum S k s d r) (N : AdviceLaw.PrivateMap data) :
AdviceExperiment.leftObservation (ActualGame.rhs S) (AdviceLaw.completeDraw data N) =
datumLeft S data := by
unfold datumLeft
rw [AdviceLaw.completeDraw_left, AdviceLaw.completeDraw_left]
theorem datumMatrix_rows (S : Source) (data : Datum S k s d r) (N : AdviceLaw.PrivateMap data) :
data.1.2.comp (datumMatrix S data N) = data.2.2.2.val :=
AdviceLaw.completeDraw_rows data N
theorem datumMatrix_rowAdvice (S : Source) (data : Datum S k s d r)
(N : AdviceLaw.PrivateMap data) :
RowErasureMatrix.rowAdvice (LinearMap.toMatrix' data.1.2)
(mapEquiv k s ((datumMatrix S data N).comp (datumProjection S data))) =
mapEquiv k r (datumLeft S data).rows := by
rw [← mapEquiv_rowAdvice]
change mapEquiv k r ((data.1.2.comp (datumMatrix S data N)).comp (datumProjection S data)) =
mapEquiv k r ((data.1.2.comp (datumMatrix S data 0)).comp (datumProjection S data))
rw [datumMatrix_rows, datumMatrix_rows]
rfl
/-- Every event membership uses the same selected normalized witness on the
entire complete-data fiber. -/
theorem slice_assemble_iff (S : Source)
(labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2 ^ s))
(α : ℝ) (hα : 0 < α) (hsmall : 1 / (2 : ℝ) ^ (s - r) < α / 8)
(data : Datum S k s d r) (hgood : GoodAdvice S labeling α (datumLeft S data))
(N : AdviceLaw.PrivateMap data) :
slice S k s d r labeling α (AdviceLaw.completeAssemble data N) = true ↔
mapEquiv k s ((datumMatrix S data N).comp (datumProjection S data)) ∈
(normalizedDescription S labeling α hα hsmall (datumLeft S data) hgood).toSlice.points := by
rw [slice_eq_selectedEvent]
change decide ((RowErasureMatrix.family s (1 + 2 * k) r).selectedEvent
(fullTable S labeling (AdviceExperiment.leftObservation (ActualGame.rhs S)
(AdviceLaw.completeDraw data N))) α (LinearMap.toMatrix' data.1.2)
(mapEquiv k s ((datumMatrix S data N).comp (datumProjection S data)))) = true ↔ _
rw [datumLeft_eq, decide_eq_true_eq]
exact SelectedDecoding.selectedEvent_iff_normalized S labeling α hα hsmall
(datumLeft S data) hgood _ (datumMatrix_rowAdvice S data N)
/-- On the selected event, the actual full-table target-hit predicate equals
projected affine equality with the very same normalized coefficient/intercept. -/
theorem target_assemble_iff (S : Source)
(labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2 ^ s))
(α : ℝ) (hα : 0 < α) (hsmall : 1 / (2 : ℝ) ^ (s - r) < α / 8)
(data : Datum S k s d r) (hgood : GoodAdvice S labeling α (datumLeft S data))
(N : AdviceLaw.PrivateMap data)
(hN : slice S k s d r labeling α (AdviceLaw.completeAssemble data N) = true) :
targetHit S k s d r labeling α (AdviceLaw.completeAssemble data N) = true ↔
TableKeysRestoration.projectedAnswer S k s d labeling (PaddedLaw.choiceMask data.2.1)
(RawPrivateTable.supported (PaddedLaw.choiceMask data.2.1) (ActualGame.names S)
data.1.1 (PaddedLaw.choiceSlots data.2.1)) ((datumMatrix S data N).prod data.2.2.1) =
datumMatrix S data N (datumProjection S data
(normalizedResponse S labeling α hα hsmall (datumLeft S data) hgood).coefficient) +
(normalizedDescription S labeling α hα hsmall (datumLeft S data) hgood).intercept := by
have hp := (slice_assemble_iff S labeling α hα hsmall data hgood N).mp hN
rw [targetHit_eq_selectedMatch]
change decide ((RowErasureMatrix.family s (1 + 2 * k) r).selectedMatch
(fullTable S labeling (AdviceExperiment.leftObservation (ActualGame.rhs S)
(AdviceLaw.completeDraw data N))) α (LinearMap.toMatrix' data.1.2)
(mapEquiv k s ((datumMatrix S data N).comp (datumProjection S data)))) = true ↔ _
rw [datumLeft_eq, decide_eq_true_eq]
refine (SelectedDecoding.selectedMatch_iff_normalized S labeling α hα hsmall
(datumLeft S data) hgood _ (datumMatrix_rowAdvice S data N)).trans ?_
have hp' : mapEquiv k s ((datumMatrix S data N).comp (datumProjection S data)) ∈
(RowErasureMatrix.family s (1 + 2 * k) r).points
(normalizedDescription S labeling α hα hsmall (datumLeft S data) hgood) := hp
simp only [hp', true_and, LinearEquiv.symm_apply_apply]
rw [show fullTable S labeling (datumLeft S data)
(mapEquiv k s ((datumMatrix S data N).comp (datumProjection S data))) =
TableKeysRestoration.projectedAnswer S k s d labeling (PaddedLaw.choiceMask data.2.1)
(RawPrivateTable.supported (PaddedLaw.choiceMask data.2.1) (ActualGame.names S)
data.1.1 (PaddedLaw.choiceSlots data.2.1)) ((datumMatrix S data N).prod data.2.2.1) from
ProjectedColumnSlice.fullTable_padded S labeling (AdviceLaw.completeDraw data 0) (datumMatrix S data N)]
rfl
/-- Positive mass supplies a supporting private coordinate in the selected
fiber; neither a hidden matrix nor good advice is an additional premise. -/
theorem exists_selected_private (S : Source)
(labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2 ^ s))
(α β : ℝ) (hβ : 0 ≤ β) (hβ' : β ≤ 1) (data : Datum S k s d r)
(positive : 0 < (law S k s d r β hβ hβ').probability
(VisibleTransfer.fiberEvent AdviceLaw.completeObserve data (slice S k s d r labeling α))) :
∃ N : AdviceLaw.PrivateMap data,
slice S k s d r labeling α (AdviceLaw.completeAssemble data N) = true := by
have hex : ∃ x : Seed S k s d r,
VisibleTransfer.fiberEvent AdviceLaw.completeObserve data (slice S k s d r labeling α) x = true := by
by_contra hn
have he : VisibleTransfer.fiberEvent AdviceLaw.completeObserve data
(slice S k s d r labeling α) = fun _ => false := by
funext x
apply Bool.eq_false_iff.mpr
intro hx
exact hn ⟨x, hx⟩
rw [he, FiniteDistribution.probability_false] at positive
exact (lt_irrefl 0) positive
obtain ⟨x, hx⟩ := hex
have hx' : AdviceLaw.completeObserve x = data ∧ slice S k s d r labeling α x = true := by
simpa only [VisibleTransfer.fiberEvent, Bool.and_eq_true, decide_eq_true_eq] using hx
let N := (AdviceLaw.completeFiberEquiv data).symm ⟨x, hx'.1⟩
have he : AdviceLaw.completeAssemble data N = x :=
congrArg Subtype.val ((AdviceLaw.completeFiberEquiv data).apply_symm_apply ⟨x, hx'.1⟩)
exact ⟨N, he.symm ▸ hx'.2⟩
theorem good_of_selected_private (S : Source)
(labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2 ^ s))
(α : ℝ) (data : Datum S k s d r) (N : AdviceLaw.PrivateMap data)
(hN : slice S k s d r labeling α (AdviceLaw.completeAssemble data N) = true) :
GoodAdvice S labeling α (datumLeft S data) := by
have hg := slice_good S k s d r labeling α (AdviceLaw.completeAssemble data N) hN
change GoodAdvice S labeling α
(AdviceExperiment.leftObservation (ActualGame.rhs S) (AdviceLaw.completeDraw data N)) at hg
rw [datumLeft_eq] at hg
exact hg
/-- The actual selected private coordinates are in bijection with the complete
projected affine slice. The supporting point is used only to state its column
equations; the canonical visible base still defines the private coordinates. -/
def selectedPrivateEquiv (S : Source)
(labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2 ^ s))
(α : ℝ) (hα : 0 < α) (hsmall : 1 / (2 : ℝ) ^ (s - r) < α / 8)
(data : Datum S k s d r) (hgood : GoodAdvice S labeling α (datumLeft S data))
(Nstar : AdviceLaw.PrivateMap data)
(hstar : slice S k s d r labeling α (AdviceLaw.completeAssemble data Nstar) = true) :
{N : AdviceLaw.PrivateMap data //
slice S k s d r labeling α (AdviceLaw.completeAssemble data N) = true} ≃
RowErasureSliceQuotient.AffineSlice data.1.2
((normalizedResponse S labeling α hα hsmall (datumLeft S data) hgood).columns.map
(datumProjection S data)) (datumMatrix S data Nstar) := by
let D := normalizedDescription S labeling α hα hsmall (datumLeft S data) hgood
let e := AdviceFibers.rowFiberEquiv data.1.2 data.2.2.2.val
(AdviceFibers.observedBase data.1.2 data.2.2.2)
(AdviceFibers.observedBase_property data.1.2 data.2.2.2)
have hs : mapEquiv k s ((datumMatrix S data Nstar).comp (datumProjection S data)) ∈ D.toSlice.points :=
(slice_assemble_iff S labeling α hα hsmall data hgood Nstar).mp hstar
have hrow (N : AdviceLaw.PrivateMap data) :
data.1.2.comp (datumMatrix S data N) = data.1.2.comp (datumMatrix S data Nstar) := by
rw [datumMatrix_rows, datumMatrix_rows]
let lift := fun M : RowErasureSliceQuotient.AffineSlice data.1.2
((normalizedResponse S labeling α hα hsmall (datumLeft S data) hgood).columns.map
(datumProjection S data)) (datumMatrix S data Nstar) =>
e.symm ⟨M.val, M.property.1.trans (datumMatrix_rows S data Nstar)⟩
have hlift (M) : datumMatrix S data (lift M) = M.val :=
congrArg Subtype.val (e.apply_symm_apply ⟨M.val, M.property.1.trans (datumMatrix_rows S data Nstar)⟩)
refine {
toFun := fun N => ⟨datumMatrix S data N.val, hrow N.val, ?_⟩
invFun := fun M => ⟨lift M, ?_⟩
left_inv := ?_
right_inv := ?_ }
· exact (ProjectedColumnSlice.contains_padded_iff D data.1.2
(normalizedDescription_spec S labeling α hα hsmall (datumLeft S data) hgood).1
(datumProjection S data) (datumMatrix S data Nstar) hs
(datumMatrix S data N.val) (hrow N.val)).mp
((slice_assemble_iff S labeling α hα hsmall data hgood N.val).mp N.property)
· apply (slice_assemble_iff S labeling α hα hsmall data hgood (lift M)).mpr
rw [hlift]
exact (ProjectedColumnSlice.contains_padded_iff D data.1.2
(normalizedDescription_spec S labeling α hα hsmall (datumLeft S data) hgood).1
(datumProjection S data) (datumMatrix S data Nstar) hs M.val M.property.1).mpr M.property.2
· intro N
apply Subtype.ext
exact e.symm_apply_apply N.val
· intro M
apply Subtype.ext
exact hlift M
@[simp] theorem selectedPrivateEquiv_val (S : Source)
(labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2 ^ s))
(α : ℝ) (hα : 0 < α) (hsmall : 1 / (2 : ℝ) ^ (s - r) < α / 8)
(data : Datum S k s d r) (hgood : GoodAdvice S labeling α (datumLeft S data))
(Nstar : AdviceLaw.PrivateMap data)
(hstar : slice S k s d r labeling α (AdviceLaw.completeAssemble data Nstar) = true)
(N : {N : AdviceLaw.PrivateMap data //
slice S k s d r labeling α (AdviceLaw.completeAssemble data N) = true}) :
(selectedPrivateEquiv S labeling α hα hsmall data hgood Nstar hstar N).val =
datumMatrix S data N.val := rfl
/-- Exact conditional target probability, with no assumed uniformity premise. -/
theorem conditional_eq_transferred (S : Source)
(labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2 ^ s))
(α : ℝ) (hα : 0 < α) (hsmall : 1 / (2 : ℝ) ^ (s - r) < α / 8)
(β : ℝ) (hβ : 0 ≤ β) (hβ' : β ≤ 1)
(data : Datum S k s d r) (hgood : GoodAdvice S labeling α (datumLeft S data))
(Nstar : AdviceLaw.PrivateMap data)
(hstar : slice S k s d r labeling α (AdviceLaw.completeAssemble data Nstar) = true)
(positive : 0 < (law S k s d r β hβ hβ').probability
(VisibleTransfer.fiberEvent AdviceLaw.completeObserve data (slice S k s d r labeling α))) :
((law S k s d r β hβ hβ').condition
(VisibleTransfer.fiberEvent AdviceLaw.completeObserve data (slice S k s d r labeling α))
positive).probability (targetHit S k s d r labeling α) =
SelectedDecoding.transferredAgreement S labeling (AdviceLaw.completeDraw data 0)
(normalizedResponse S labeling α hα hsmall (datumLeft S data) hgood)
(normalizedDescription S labeling α hα hsmall (datumLeft S data) hgood).intercept
(datumMatrix S data Nstar) := by
have enumeration : ActualAdviceUpper.publicMapFintype s r =
(ActualConditionalLaw.homFintype : Fintype (Alphabet s →ₗ[F2] Vector r)) :=
Subsingleton.elim _ _
unfold law at positive ⊢
revert positive
rw [enumeration]
intro positive
refine (ActualConditionalLaw.probability_condition_complete
(k := k) (Id := Fin S.occurrences) (K := Alphabet s) (W := Vector d) (R := Vector r)
(FiniteDistribution.uniform (ActualGame.Question S k))
(@FiniteDistribution.uniform (Alphabet s →ₗ[F2] Vector r)
ActualConditionalLaw.homFintype inferInstance) β hβ hβ' data
(slice S k s d r labeling α) (targetHit S k s d r labeling α) positive).trans ?_
let : Fintype (RawPartnerTarget.RawPoint (PaddedLaw.choiceMask data.2.1) →ₗ[F2] Alphabet s) :=
homFintype
let : Fintype (RowErasureSliceQuotient.AffineSlice data.1.2
((normalizedResponse S labeling α hα hsmall (datumLeft S data) hgood).columns.map
(datumProjection S data)) (datumMatrix S data Nstar)) :=
SelectedDecoding.affineSliceFintype _ _ _
let f : {N : AdviceLaw.PrivateMap data //
slice S k s d r labeling α (AdviceLaw.completeAssemble data N) = true} → ℝ :=
fun N => if targetHit S k s d r labeling α (AdviceLaw.completeAssemble data N.val) = true
then 1 else 0
let g : RowErasureSliceQuotient.AffineSlice data.1.2
((normalizedResponse S labeling α hα hsmall (datumLeft S data) hgood).columns.map
(datumProjection S data)) (datumMatrix S data Nstar) → ℝ :=
fun M => if TableKeysRestoration.projectedAnswer S k s d labeling
(PaddedLaw.choiceMask data.2.1)
(RawPrivateTable.supported (PaddedLaw.choiceMask data.2.1) (ActualGame.names S)
data.1.1 (PaddedLaw.choiceSlots data.2.1)) (M.val.prod data.2.2.1) =
M.val (datumProjection S data
(normalizedResponse S labeling α hα hsmall (datumLeft S data) hgood).coefficient) +
(normalizedDescription S labeling α hα hsmall (datumLeft S data) hgood).intercept
then 1 else 0
have he : (𝔼 N, f N) = 𝔼 M, g M := by
refine Fintype.expect_equiv
(selectedPrivateEquiv S labeling α hα hsmall data hgood Nstar hstar) f g ?_
intro N
dsimp only [f, g]
simp only [selectedPrivateEquiv_val]
exact TransferredDecoding.indicator_congr _ _ _ _
(target_assemble_iff S labeling α hα hsmall data hgood N.val N.property)
calc
_ = 𝔼 N, f N := by
apply Finset.expect_congr (by ext N; simp only [Finset.mem_univ])
intro N _
exact TransferredDecoding.indicator_congr _ _ _ _ Iff.rfl
_ = 𝔼 M, g M := he
_ = _ := by
unfold SelectedDecoding.transferredAgreement
apply Finset.expect_congr (univ_congr _ _)
intro M _
exact TransferredDecoding.indicator_congr _ _ _ _ Iff.rfl
/-- Every complete visible datum selected by the actual witness event pays the
private-decoding success constant. Good advice and a nonempty transferred
slice follow from positivity and are not additional hypotheses. -/
theorem visibleWitness_decoding (S : Source)
(labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2 ^ s))
(α : ℝ) (hα : 0 < α) (hsmall : 1 / (2 : ℝ) ^ (s - r) < α / 8)
(β : ℝ) (hβ : 0 ≤ β) (hβ' : β ≤ 1) (data : Datum S k s d r)
(hvisible : VisibleTransfer.visibleWitness (law S k s d r β hβ hβ')
AdviceLaw.completeObserve (slice S k s d r labeling α)
(targetHit S k s d r labeling α) (α / 4) data = true) :
(α / 8) ^ 2 / (2 : ℝ) ^ (s * r) / (2 : ℝ) ^ r ≤
ActualConditionalLaw.datumAgreement S labeling (goodPredicate S labeling α hα hsmall) data := by
obtain ⟨positive, hagree⟩ := VisibleTransfer.visibleWitness_conditional
(law S k s d r β hβ hβ') AdviceLaw.completeObserve
(slice S k s d r labeling α) (targetHit S k s d r labeling α) (α / 4) data hvisible
obtain ⟨Nstar, hstar⟩ := exists_selected_private S labeling α β hβ hβ' data positive
have hgood := good_of_selected_private S labeling α data Nstar hstar
rw [conditional_eq_transferred S labeling α hα hsmall β hβ hβ' data hgood Nstar hstar positive] at hagree
apply SelectedDecoding.observedAgreement_lower S labeling α hα hsmall
(AdviceLaw.completeDraw data 0) hgood (datumMatrix S data Nstar) ?_ hagree
exact (datumMatrix_rows S data Nstar).trans (datumMatrix_rows S data 0).symm
end UniqueGamesTheorem.Decoder.SelectedFiberLaw
end
end OAI
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