| import OAI.Geometry.PinnedDistances.FiniteObstruction |
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| namespace OAI |
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| universe u1 u2 |
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| 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)] |
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| /-- 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 |
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| end WeakPinned.Bounded |
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| end |
| end |
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| end OAI |
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