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import OAI.Computability.VertexCover.Analysis.SeparatorBoundCube
import OAI.Computability.VertexCover.Reduction.PrivateProjectionGroupLower
namespace OAI
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namespace VertexCover.LabelCover
open MeasureTheory
open VertexCover.Restriction
open EnergyForm.Projection
open scoped BigOperators
theorem spliceSeeds_at_inside (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
(frozen : Φ.Seeds d) (hidden : Φ.HiddenSeeds J) (e : ↥(internalPairs J)) :
Φ.spliceSeeds J frozen hidden e = hidden e := by
simp only [spliceSeeds, dite_eq_left e.2]
theorem spliceSeeds_at_outside (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
(frozen : Φ.Seeds d) (hidden : Φ.HiddenSeeds J) (e : PositionPair d)
(he : e ∉ internalPairs J) : Φ.spliceSeeds J frozen hidden e = frozen e := by
simp only [spliceSeeds, dite_eq_right he]
noncomputable def privateFiber (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
(j : Fin d) (seed : Φ.Seeds d) : Finset (Φ.Seeds d) := by
classical
exact Finset.univ.filter (fun t => Φ.privateSeedKey J j t = Φ.privateSeedKey J j seed)
@[simp] theorem mem_privateFiber (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
(j : Fin d) (seed t : Φ.Seeds d) : t ∈ Φ.privateFiber J j seed ↔
Φ.privateSeedKey J j t = Φ.privateSeedKey J j seed := by
classical
simp only [privateFiber, Finset.mem_filter, Finset.mem_univ, true_and]
theorem spliceSeeds_restrict_of_outside (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
(frozen seed : Φ.Seeds d) (ho : ∀ e, e ∉ internalPairs J → frozen e = seed e) :
Φ.spliceSeeds J frozen (fun e : ↥(internalPairs J) => seed e) = seed := by
funext e
by_cases he : e ∈ internalPairs J
· simp only [spliceSeeds, dite_eq_left he]
· simpa only [spliceSeeds, dite_eq_right he] using ho e he
noncomputable def privateFiberEquiv (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
(frozen : Φ.Seeds d) (j : J) (seed : Φ.Seeds d)
(ho : ∀ e, e ∉ internalPairs J → frozen e = seed e) :
Φ.ownFiber J frozen j (Φ.query seed j) ≃ Φ.privateFiber J j seed := by
classical
have reconstruct : ∀ t : Φ.privateFiber J j seed,
Φ.spliceSeeds J frozen (fun e : ↥(internalPairs J) => t.1 e) = t.1 := by
intro t
apply Φ.spliceSeeds_restrict_of_outside
intro e he
have ht := (Φ.privateSeedKey_eq_iff J j t.1 seed).mp ((Φ.mem_privateFiber J j seed t.1).mp t.2)
exact (ho e he).trans (ht.1 e he).symm
exact {
toFun := fun hidden => ⟨Φ.spliceSeeds J frozen hidden.1, by
apply (Φ.mem_privateFiber J j seed _).mpr
apply (Φ.privateSeedKey_eq_iff J j _ _).mpr
exact ⟨fun e he => (Φ.spliceSeeds_at_outside J frozen hidden.1 e he).trans (ho e he),
(Φ.mem_ownFiber J frozen j (Φ.query seed j) hidden.1).mp hidden.2⟩⟩
invFun := fun t => ⟨fun e => t.1 e, by
apply (Φ.mem_ownFiber J frozen j (Φ.query seed j) _).mpr
rw [reconstruct t]
exact ((Φ.privateSeedKey_eq_iff J j t.1 seed).mp
((Φ.mem_privateFiber J j seed t.1).mp t.2)).2⟩
left_inv := fun hidden => by
apply Subtype.ext
funext e
exact Φ.spliceSeeds_at_inside J frozen hidden.1 e
right_inv := fun t => Subtype.ext (reconstruct t) }
theorem fiberAverage_eq_finiteMean {S T : Type*} [Fintype S]
(q : S → T) (f : S → ℝ) (i : T) (ts : Finset S)
(ht : ∀ s, s ∈ ts ↔ q s = i) :
VertexCover.Average.fiberAverage q (fun _ => f) i =
VertexCover.finiteMean (fun s : ts => f s) := by
classical
have he : Finset.univ.filter (fun s => q s = i) = ts := by
ext s
simpa only [Finset.mem_filter, Finset.mem_univ, true_and] using (ht s).symm
unfold VertexCover.Average.fiberAverage
rw [he, VertexCover.Average.finiteMean_finset ts f]
theorem privateSeedProjection_fiber (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
(frozen : Φ.Seeds d) (j : J) (seed : Φ.Seeds d)
(ho : ∀ e, e ∉ internalPairs J → frozen e = seed e)
(f : Φ.RestrictionSpace d) (s : Φ.RestrictionWeights d) :
Φ.privateSeedProjection J j f seed s =
VertexCover.finiteMean (fun hidden : Φ.ownFiber J frozen j (Φ.query seed j) =>
f (Φ.spliceSeeds J frozen hidden.1) s) := by
classical
change conditionMap _ f seed s = _
rw [conditionMap_apply_function]
calc
_ = VertexCover.finiteMean (fun t : Φ.privateFiber J j seed => f t.1 s) := by
exact fiberAverage_eq_finiteMean (Φ.privateSeedKey J j) (fun t => f t s)
(Φ.privateSeedKey J j seed) (Φ.privateFiber J j seed)
(fun t => Φ.mem_privateFiber J j seed t)
_ = _ := (VertexCover.finiteMean_equiv (Φ.privateFiberEquiv J frozen j seed ho)
(fun t : Φ.privateFiber J j seed => f t.1 s)).symm
theorem otherWeightProjection_apply (Φ : LabelCover) {d : ℕ}
(J : Finset (Fin d)) (j : J) (f : Φ.RestrictionSpace d)
(seed : Φ.Seeds d) (s : Φ.RestrictionWeights d) :
Φ.otherWeightProjection J j f seed s =
∫ t, f seed (splice J s (Function.update t j (s j)))
∂CompactCube.blockLaw J (Fin (Φ.WeightDimension d)) := by
change productMap (fun i => (average (CompactCube.law (Fin (Φ.WeightDimension d))) i).lift)
(J.erase j).toList f seed s = _
rw [productMap_lift, product_average_apply _ _ (J.erase j).nodup_toList,
Finset.toList_toFinset]
exact integral_plug_erase _ J j (f seed) s
theorem realWeights_splice (Φ : LabelCover) {d : ℕ}
(J : Finset (Fin d)) (s : Φ.RestrictionWeights d)
(t : J → Fin (Φ.WeightDimension d) → CompactCube.Interval) :
CompactCube.realWeights (splice J s t) =
Φ.spliceWeights J (CompactCube.realWeights s) (CompactCube.realWeights t) := by
classical
funext i k
by_cases hi : i ∈ J
· simp only [splice, spliceWeights, CompactCube.realWeights, dite_eq_left hi]
· simp only [splice, spliceWeights, CompactCube.realWeights, dite_eq_right hi]
theorem otherWeightProjection_separator_splice (Φ : LabelCover) {d : ℕ}
(J : Finset (Fin d)) (frozen : Φ.Seeds d) (j : J)
(A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty)
(hidden : Φ.HiddenSeeds J) (s : Φ.RestrictionWeights d) :
Φ.otherWeightProjection J j (Φ.restrictionFunction A hA)
(Φ.spliceSeeds J frozen hidden) s =
∫ other, Φ.batchFunction (Φ.spliceSeeds J frozen hidden) J
(Φ.outsideSum J frozen (CompactCube.realWeights s)) A hA
(Function.update other j (CompactCube.realWeights s j)) ∂Φ.batchLaw J := by
classical
rw [Φ.otherWeightProjection_apply]
simp only [restrictionFunction_apply, Φ.realWeights_splice, CompactCube.realWeights_update]
simp_rw [← Φ.batchFunction_is_restriction J frozen (CompactCube.realWeights s) A hA]
apply CompactCube.integral_realWeights (f := fun other : Φ.BatchWeights J =>
Φ.batchFunction (Φ.spliceSeeds J frozen hidden) J
(Φ.outsideSum J frozen (CompactCube.realWeights s)) A hA
(Function.update other j (CompactCube.realWeights s j)))
exact ((Φ.batchFunction_continuous _ J _ A hA).comp
(continuous_id.update j continuous_const)).measurable.aestronglyMeasurable
theorem privateAverageProjection_separator (Φ : LabelCover) {d : ℕ}
(J : Finset (Fin d)) (frozen : Φ.Seeds d) (j : J) (seed : Φ.Seeds d)
(ho : ∀ e, e ∉ internalPairs J → frozen e = seed e)
(A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) (s : Φ.RestrictionWeights d) :
Φ.privateAverageProjection J j (Φ.restrictionFunction A hA) seed s =
Φ.ownMean J frozen (Φ.outsideSum J frozen (CompactCube.realWeights s)) A hA
j (Φ.query seed j) (CompactCube.realWeights s j) := by
change Φ.privateSeedProjection J j
(Φ.otherWeightProjection J j (Φ.restrictionFunction A hA)) seed s = _
rw [Φ.privateSeedProjection_fiber J frozen j seed ho]
unfold ownMean
congr 1
funext hidden
exact Φ.otherWeightProjection_separator_splice J frozen j A hA hidden.1 s
theorem outsideSum_update (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
(frozen : Φ.Seeds d) (s : Fin d → Fin (Φ.WeightDimension d) → ℝ) (j : J)
(x : Fin (Φ.WeightDimension d) → ℝ) :
Φ.outsideSum J frozen (Function.update s j x) = Φ.outsideSum J frozen s := by
classical
apply Finset.sum_congr rfl
intro k hk
have hkj : k ≠ j := by
intro h
exact (Finset.mem_compl.mp hk) (h ▸ j.2)
rw [Function.update_of_ne hkj]
theorem privateProjection_separator (Φ : LabelCover) {d : ℕ}
(J : Finset (Fin d)) (frozen : Φ.Seeds d) (j : J) (seed : Φ.Seeds d)
(ho : ∀ e, e ∉ internalPairs J → frozen e = seed e)
(A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) (s : Φ.RestrictionWeights d) :
Φ.privateProjection J j (Φ.restrictionFunction A hA) seed s =
Φ.ownMean J frozen (Φ.outsideSum J frozen (CompactCube.realWeights s)) A hA
j (Φ.query seed j) (CompactCube.realWeights s j) -
∫ x, Φ.ownMean J frozen (Φ.outsideSum J frozen (CompactCube.realWeights s)) A hA
j (Φ.query seed j) x ∂VertexCover.Cube.law (Fin (Φ.WeightDimension d)) := by
classical
change Φ.privateAverageProjection J j (Φ.restrictionFunction A hA) seed s -
Φ.weightProjection j (Φ.privateAverageProjection J j (Φ.restrictionFunction A hA)) seed s = _
rw [Φ.privateAverageProjection_separator J frozen j seed ho,
Φ.weightProjection_apply]
simp_rw [Φ.privateAverageProjection_separator J frozen j seed ho,
CompactCube.realWeights_update, Φ.outsideSum_update J frozen _ j,
Function.update_self]
congr 1
exact CompactCube.integral_coe
(Φ.ownMean_lipschitz J frozen _ A hA j (Φ.query seed j)).continuous.measurable.aestronglyMeasurable
theorem privateProjection_coordinate_variance (Φ : LabelCover) {d : ℕ}
(J : Finset (Fin d)) (frozen : Φ.Seeds d) (j : J) (seed : Φ.Seeds d)
(ho : ∀ e, e ∉ internalPairs J → frozen e = seed e)
(A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) (s : Φ.RestrictionWeights d) :
(∫ x, (Φ.privateProjection J j (Φ.restrictionFunction A hA) seed
(Function.update s j x))^2 ∂CompactCube.law (Fin (Φ.WeightDimension d))) =
ProbabilityTheory.variance (Φ.ownMean J frozen
(Φ.outsideSum J frozen (CompactCube.realWeights s)) A hA j (Φ.query seed j))
(VertexCover.Cube.law (Fin (Φ.WeightDimension d))) := by
classical
simp_rw [Φ.privateProjection_separator J frozen j seed ho,
CompactCube.realWeights_update, Φ.outsideSum_update J frozen _ j,
Function.update_self]
rw [ProbabilityTheory.variance_eq_integral
(Φ.ownMean_lipschitz J frozen _ A hA j (Φ.query seed j)).continuous.measurable.aemeasurable]
apply CompactCube.integral_coe (f := fun x =>
(Φ.ownMean J frozen (Φ.outsideSum J frozen (CompactCube.realWeights s)) A hA j (Φ.query seed j) x -
∫ y, Φ.ownMean J frozen (Φ.outsideSum J frozen (CompactCube.realWeights s)) A hA j (Φ.query seed j) y
∂VertexCover.Cube.law (Fin (Φ.WeightDimension d)))^2)
exact (((Φ.ownMean_lipschitz J frozen _ A hA j (Φ.query seed j)).continuous.sub
continuous_const).pow 2).measurable.aestronglyMeasurable
end VertexCover.LabelCover
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end OAI
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