File size: 4,207 Bytes
2fca478 | 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 | import OAI.Geometry.PinnedDistances.FiniteObstruction
namespace OAI
universe u1 u2
section
noncomputable section
open MeasureTheory Filter Set
open scoped Topology
namespace WeakPinned.Bounded
local instance : MeasurableSpace Sphere := borel Sphere
local instance : BorelSpace Sphere := ⟨rfl⟩
variable {V : ℕ → Type u1} {I : ℕ → Type u2} [∀ n,Fintype (V n)] [∀ n,DecidableEq (V n)] [∀ n,Fintype (I n)]
/-- The actual finite expanded graph laws cannot have nonatomic coordinate limits
and the two test-independence properties supplied by the variance estimates. -/
theorem graph_limit_impossible
(G : ∀ n,Core.Fibers (V n) (I n)) (U₁ U₂ : ∀ n,V n → ℂ)
(hG : ∀ n,0<(G n).edgeCount) (hB : ∀ n e,((G n).targets e).Nonempty)
(hn : ∀ n,2≤Fintype.card (V n))
(β₁ β₂ : ProbabilityMeasure Sphere) (Q : ProbabilityMeasure (Quad Sphere))
(R : ProbabilityMeasure (Expanded Sphere))
[NullSingletonClass (β₁:Measure Sphere)] [NullSingletonClass (β₂:Measure Sphere)]
(hα₁ : Tendsto (fun n => uniformProbability (fun x => (U₁ n x:Sphere)) (by have:=hn n;omega)) atTop (𝓝 β₁))
(hα₂ : Tendsto (fun n => uniformProbability (fun x => (U₂ n x:Sphere)) (by have:=hn n;omega)) atTop (𝓝 β₂))
(hQ : Tendsto (fun n => edgeProbability (G n) (fun x => (U₁ n x:Sphere))
(fun x => (U₂ n x:Sphere)) (hG n) (hB n)) atTop (𝓝 Q))
(hR : Tendsto (fun n => expandedProbability (G n) (fun x => (U₁ n x:Sphere))
(fun x => (U₂ n x:Sphere)) (hG n) (hB n)) atTop (𝓝 R))
(hd₁ : ∀ f : C(Sphere,ℝ),Tendsto (fun n => testDiscrep (G n).fiberWeight (G n).targets
(fun x => f (U₁ n x:Sphere))) atTop (𝓝 0))
(hd₂ : ∀ f : C(Sphere,ℝ),Tendsto (fun n => testDiscrep (G n).fiberWeight (G n).targets
(fun x => f (U₂ n x:Sphere))) atTop (𝓝 0))
(T : ℕ → ℝ) (hT : ∀ n,0<T n)
(hdR : ∀ f : C(Sphere,ℝ),Tendsto (fun n => testDiscrep ((G n).retainedWeight (T n))
((G n).retainedSet (T n)) (fun x => f (U₁ n x:Sphere))) atTop (𝓝 0))
(ho : Tendsto (fun n => (G n).omittedMass (T n)) atTop (𝓝 0))
(M : ℝ) (hM : 0≤M) (hdom : ∀ n x y,(G n).edgeLaw x y≤M*Variance.pairProb Finset.univ x y)
(hrel : ∀ n e y z,y∈(G n).targets e → z∈(G n).targets e →
(U₁ n y-U₁ n ((G n).source e))*(U₂ n y-U₂ n ((G n).source e))=
(U₁ n z-U₁ n ((G n).source e))*(U₂ n z-U₂ n ((G n).source e))) : False := by
let : TopologicalSpace.PseudoMetrizableSpace Sphere := inferInstance
let : TopologicalSpace.PseudoMetrizableSpace (Quad Sphere) := inferInstance
let : TopologicalSpace.PseudoMetrizableSpace (Sphere × Sphere) := inferInstance
let : TopologicalSpace.PseudoMetrizableSpace (Expanded Sphere) := inferInstance
let S₁ := fun n x => (U₁ n x:Sphere)
let S₂ := fun n x => (U₂ n x:Sphere)
have hm₁ := expanded_marginal_limit G S₁ S₂ hG hB hn Prod.fst continuous_fst β₁ Q R hα₁ hQ hR hd₁
have hm₂ := expanded_marginal_limit G S₁ S₂ hG hB hn Prod.snd continuous_snd β₂ Q R hα₂ hQ hR hd₂
have he := weak_map_eq
(fun n => expandedProbability (G n) (S₁ n) (S₂ n) (hG n) (hB n)) R
(fun n => edgeProbability (G n) (S₁ n) (S₂ n) (hG n) (hB n)) Q
Prod.fst continuous_fst hR hQ
(fun n => expandedProbability_edge (G n) (S₁ n) (S₂ n) (hG n) (hB n))
have hi := incoming_limit G S₁ S₂ hG hB hn T hT β₁ Q hα₁ hQ hdR ho
have hac := edge_ac_limit G S₁ S₂ hG hB hn β₁ β₂ Q hα₁ hα₂ hQ M hM hdom
have hz := weak_chart_closed expandedCoe expandedCoe_open
(fun n => expandedProbability (G n) (S₁ n) (S₂ n) (hG n) (hB n)) R hR
zEquation zEquation_closed
(fun n => (expanded_chart_zero (G n) (U₁ n) (U₂ n) (hG n) (hB n) (hrel n)).1)
have hw := weak_chart_closed expandedCoe expandedCoe_open
(fun n => expandedProbability (G n) (S₁ n) (S₂ n) (hG n) (hB n)) R hR
wEquation wEquation_closed
(fun n => (expanded_chart_zero (G n) (U₁ n) (U₂ n) (hG n) (hB n) (hrel n)).2)
exact no_sphere_expanded β₁ β₂ Q R he hm₁ hm₂ hi hac.1 hac.2 hz hw
end WeakPinned.Bounded
end
end
end OAI
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