| import OAI.Geometry.PinnedDistances.PartitionIntegrals |
|
|
| namespace OAI |
|
|
| universe u1 u2 u3 u4 u5 u6 u7 u8 u9 u10 u11 |
|
|
| section |
| noncomputable section |
| open scoped BigOperators Topology ZeroAtInfty |
| open Filter MeasureTheory |
| namespace WeakPinned.Bounded |
|
|
| lemma test_bound_rate (D V A : ℕ → ℝ) (H M : ℝ) |
| (hD : ∀ n,0≤ D n) (hH : 0≤ H) |
| (hV : Tendsto V atTop (𝓝 0)) (hA : Tendsto A atTop (𝓝 0)) |
| (h : ∀ η,0<η → ∃ N : ℝ,0≤ N ∧ ∀ δ,0<δ → |
| ∀ n,D n≤ 2*η+H*(V n/δ+N*(δ+(M+1)*A n))) : |
| Tendsto D atTop (𝓝 0) := by |
| apply tendsto_order.mpr |
| constructor |
| · intro a ha |
| exact Eventually.of_forall (fun n => ha.trans_le (hD n)) |
| · intro b hb |
| obtain ⟨N,hN,h⟩ := h (b/8) (by positivity) |
| let δ := b/(4*(H*N+1)) |
| have hδ : 0<δ := by dsimp [δ]; positivity |
| have he : (H*N+1)*δ=b/4 := by dsimp [δ]; field_simp |
| have hNd : H*N*δ≤ b/4 := by nlinarith |
| have hlt : 2*(b/8)+H*(0/δ+N*(δ+(M+1)*0))<b := by simp only [zero_div,mul_zero,add_zero]; nlinarith |
| have ht := (tendsto_const_nhds.add ((hV.div_const δ).add |
| ((tendsto_const_nhds.add (hA.const_mul (M+1))).const_mul N) |>.const_mul H)).eventually |
| (eventually_lt_nhds hlt) |
| exact ht.mono (fun n hn => (h δ hδ n).trans_lt hn) |
|
|
| lemma zeroAtInfty_tail (f : C₀(ℂ,ℝ)) (η : ℝ) (hη : 0<η) : |
| ∃ R : ℝ,0≤ R ∧ ∀ z,R<‖z‖ → |f z|≤ η := by |
| have hf := (Metric.tendsto_nhds.mp (zero_at_infty f) η hη) |
| obtain ⟨K,hK,hsub⟩ := mem_cocompact.mp hf |
| obtain ⟨R,hR⟩ := hK.isBounded.subset_closedBall (0:ℂ) |
| refine ⟨max R 0,le_max_right _ _,?_⟩ |
| intro z hz |
| have hzK : z∉K := by |
| intro hzK |
| have hh := Metric.mem_closedBall.mp (hR hzK) |
| simp only [dist_zero_right] at hh |
| exact (not_le_of_gt (lt_of_le_of_lt (le_max_left _ _) hz)) hh |
| simpa only [Real.dist_eq,sub_zero] using (show dist (f z) 0<η from hsub hzK).le |
|
|
| lemma zeroAtInfty_modulus (f : C₀(ℂ,ℝ)) (η : ℝ) (hη : 0<η) : |
| ∃ ε : ℝ,0<ε ∧ ∀ z w,‖z-w‖<2*ε → |f z-f w|≤ η := by |
| obtain ⟨δ,hδ,hd⟩ := Metric.uniformContinuous_iff.mp (ZeroAtInftyContinuousMap.uniformContinuous f) η hη |
| refine ⟨δ/2,by positivity,?_⟩ |
| intro z w hzw |
| have hh := hd (show dist z w<δ by rw [dist_eq_norm]; linarith) |
| exact (by simpa only [Real.dist_eq] using hh.le) |
|
|
| local instance : MeasurableSpace Sphere := borel Sphere |
| local instance : BorelSpace Sphere := ⟨rfl⟩ |
|
|
| def sphereTest (f : C(Sphere,ℝ)) : C₀(ℂ,ℝ) where |
| toFun := fun z => f (z:Sphere)-f OnePoint.infty |
| continuous_toFun := (f.continuous.comp OnePoint.continuous_coe).sub continuous_const |
| zero_at_infty' := by |
| have hco : Tendsto ((↑) : ℂ → Sphere) (cocompact ℂ) (𝓝 OnePoint.infty) := |
| OnePoint.tendsto_coe_infty.mono_left cocompact_le_coclosedCompact |
| simpa using ((f.continuous.tendsto OnePoint.infty).comp hco).sub_const (f OnePoint.infty) |
|
|
| lemma sphereTest_bound (f : C(Sphere,ℝ)) (z : ℂ) : |sphereTest f z|≤ 2*‖f‖ := by |
| have h₁ := f.norm_coe_le_norm (z:Sphere) |
| have h₂ := f.norm_coe_le_norm OnePoint.infty |
| change |f (z:Sphere)-f OnePoint.infty|≤ 2*‖f‖ |
| simpa only [Real.norm_eq_abs,two_mul] using |
| (norm_sub_le (f (z:Sphere)) (f OnePoint.infty)).trans (add_le_add h₁ h₂) |
|
|
| end WeakPinned.Bounded |
|
|
| end |
| end |
| section |
| noncomputable section |
| open scoped BigOperators Topology ZeroAtInfty |
| open Filter MeasureTheory |
| namespace WeakPinned.Bounded |
| open Hierarchy Variance Global NumberField |
| variable {P : Type u1} {I : Type u2} [Fintype P] [DecidableEq P] [Fintype I] |
|
|
| lemma testDiscrep_sub_const (w : I → ℝ) (B : I → Finset P) (f : P → ℝ) (c : ℝ) |
| (hn : 0<Fintype.card P) (hB : ∀ i,(B i).Nonempty) : |
| testDiscrep w B (fun x => f x-c)=testDiscrep w B f := by |
| have hh (S : Finset P) (hS : S.Nonempty) : |
| (∑ x,prob S x*(f x-c))=(∑ x,prob S x*f x)-c := by |
| simp_rw [mul_sub] |
| rw [Finset.sum_sub_distrib,← Finset.sum_mul,sum_prob S hS,one_mul] |
| unfold testDiscrep |
| simp_rw [hh _ (hB _),hh Finset.univ (Finset.univ_nonempty_iff.mpr (Fintype.card_pos_iff.mp hn))] |
| simp only [sub_sub_sub_cancel_right] |
|
|
| variable {V : ℕ → Type u3} {J : ℕ → Type u4} {K : ℕ → Type u5} |
| [∀ n,Fintype (V n)] [∀ n,DecidableEq (V n)] [∀ n,Fintype (J n)] |
| [∀ n,Field (K n)] [∀ n,NumberField (K n)] |
|
|
| omit [∀ (n : ℕ), NumberField (K n)] in |
| lemma C0_arch_test_rate (Z₁ Z₂ : ∀ n,V n → K n) |
| (h₁ : ∀ n,Function.Injective (Z₁ n)) (h₂ : ∀ n,Function.Injective (Z₂ n)) |
| (v : ∀ n,InfinitePlace (K n)) (m : ℕ → ℝ) (b : ℕ → ℂ) |
| (w : ∀ n,J n → ℝ) (B : ∀ n,J n → Finset (V n)) (T : ℕ → ℝ) (M H : ℝ) |
| (hn : ∀ n,2≤ Fintype.card (V n)) (hT : ∀ n,1≤ T n) (hM : 0≤ M) (hH : 0≤ H) |
| (hw : ∀ n i,0≤ w n i) (hW : ∀ n,∑ i,w n i≤ 1) (hB : ∀ n i,2≤ (B n i).card) |
| (hdom : ∀ n x,(∑ i,w n i*prob (B n i) x)≤ M*prob Finset.univ x) |
| (hvar : Tendsto (fun n => archMean (Z₁ n) (Z₂ n) (massEval (varianceCell (w n) (B n) (T n))) (v n)) atTop (𝓝 0)) |
| (hsmall : Tendsto (fun n => T n/Fintype.card (V n)) atTop (𝓝 0)) |
| (f : C₀(ℂ,ℝ)) (hbound : ∀ z,|f z|≤ H) : |
| Tendsto (fun n => testDiscrep (w n) (B n) |
| (fun x => f ((Real.exp (m n):ℂ)*(v n).embedding (Z₁ n x)+b n))) atTop (𝓝 0) := by |
| apply test_bound_rate _ _ _ H M (fun n => testDiscrep_nonneg _ _ _ (hw n)) hH hvar hsmall |
| intro η hη |
| obtain ⟨R,hR,hout⟩ := zeroAtInfty_tail f η hη |
| obtain ⟨ε,hε,hcont⟩ := zeroAtInfty_modulus f η hη |
| obtain ⟨N,hN⟩ := exists_nat_ge (2*R/(ε/(2*Real.exp 1))+1) |
| refine ⟨((2*N+1)^2:ℕ),by positivity,?_⟩ |
| intro δ hδ n |
| have hh := arch_test_bound (Z₁ n) (Z₂ n) (h₁ n) (h₂ n) (v n) (m n) (b n) (hn n) |
| R ε η H M (T n) δ N hR hε hN hη.le hH hM (hT n) hδ |
| (w n) (B n) (hw n) (hW n) (hB n) (hdom n) f hbound hcont hout |
| convert hh using 1 |
| ring |
|
|
| local instance : MeasurableSpace Sphere := borel Sphere |
| local instance : BorelSpace Sphere := ⟨rfl⟩ |
|
|
| omit [∀ (n : ℕ), NumberField (K n)] in |
| lemma sphere_arch_test_rate (Z₁ Z₂ : ∀ n,V n → K n) |
| (h₁ : ∀ n,Function.Injective (Z₁ n)) (h₂ : ∀ n,Function.Injective (Z₂ n)) |
| (v : ∀ n,InfinitePlace (K n)) (m : ℕ → ℝ) (b : ℕ → ℂ) |
| (w : ∀ n,J n → ℝ) (B : ∀ n,J n → Finset (V n)) (T : ℕ → ℝ) (M : ℝ) |
| (hn : ∀ n,2≤ Fintype.card (V n)) (hT : ∀ n,1≤ T n) (hM : 0≤ M) |
| (hw : ∀ n i,0≤ w n i) (hW : ∀ n,∑ i,w n i≤ 1) (hB : ∀ n i,2≤ (B n i).card) |
| (hdom : ∀ n x,(∑ i,w n i*prob (B n i) x)≤ M*prob Finset.univ x) |
| (hvar : Tendsto (fun n => archMean (Z₁ n) (Z₂ n) (massEval (varianceCell (w n) (B n) (T n))) (v n)) atTop (𝓝 0)) |
| (hsmall : Tendsto (fun n => T n/Fintype.card (V n)) atTop (𝓝 0)) |
| (f : C(Sphere,ℝ)) : |
| Tendsto (fun n => testDiscrep (w n) (B n) |
| (fun x => f (((Real.exp (m n):ℂ)*(v n).embedding (Z₁ n x)+b n):Sphere))) atTop (𝓝 0) := by |
| have ht := C0_arch_test_rate Z₁ Z₂ h₁ h₂ v m b w B T M (2*‖f‖) hn hT hM (by positivity) |
| hw hW hB hdom hvar hsmall (sphereTest f) (sphereTest_bound f) |
| convert ht using 1 |
| funext n |
| symm |
| apply testDiscrep_sub_const _ _ _ _ (by have := hn n; omega) |
| intro i |
| exact Finset.card_pos.mp (by have := hB n i; omega) |
|
|
| end WeakPinned.Bounded |
|
|
| end |
| end |
| section |
| noncomputable section |
| open MeasureTheory Filter |
| open scoped BigOperators Topology |
| namespace WeakPinned.Bounded |
| open Variance |
| variable {V : Type u6} {I : Type u7} [Fintype V] [DecidableEq V] [Fintype I] |
| variable (G : Core.Fibers V I) |
|
|
| omit [Fintype I] in |
| lemma incoming_discrep_bound (T H : ℝ) (hT : 0<T) (hn : 2≤Fintype.card V) |
| (f : V → ℝ) (hf : ∀ x,|f x|≤H) : |
| (∑ y,G.incomingWeight y*|(∑ x,G.conditionalSource y x*f x)-∑ x,prob Finset.univ x*f x|)≤ |
| testDiscrep (G.retainedWeight T) (G.retainedSet T) f+2*H*G.omittedMass T := by |
| have hA : |∑ x,prob (Finset.univ : Finset V) x*f x|≤H := |
| avg_abs_bound _ _ _ (Core.Fibers.prob_nn _) (sum_prob _ |
| (Finset.univ_nonempty_iff.mpr (Fintype.card_pos_iff.mp (by omega)))) hf |
| unfold testDiscrep Core.Fibers.omittedMass |
| rw [Finset.mul_sum,← Finset.sum_add_distrib] |
| apply Finset.sum_le_sum |
| intro y _ |
| by_cases hy : T≤((G.incoming y).card:ℝ) |
| · have hne : (G.incoming y).Nonempty := Finset.card_pos.mp (by exact_mod_cast hT.trans_le hy) |
| simp only [Core.Fibers.retainedWeight,Core.Fibers.retainedSet,Core.Fibers.conditionalSource, |
| ite_eq_left hy,ite_eq_left hne,not_lt_of_ge hy,ite_false,mul_zero,add_zero] |
| exact le_rfl |
| · have hbound : |(∑ x,G.conditionalSource y x*f x)-∑ x,prob Finset.univ x*f x|≤2*H := by |
| have hh := avg_abs_bound _ _ _ (G.conditionalSource_nonneg y) (G.conditionalSource_sum hn y) hf |
| exact (abs_sub _ _).trans (by linarith) |
| simp only [Core.Fibers.retainedWeight,ite_eq_right hy,lt_of_not_ge hy,ite_true,zero_mul,zero_add] |
| nlinarith [mul_le_mul_of_nonneg_left hbound (G.incomingWeight_nonneg y)] |
|
|
| lemma incoming_test_bound (T H C : ℝ) (hT : 0<T) (hn : 2≤Fintype.card V) |
| (f g : V → ℝ) (hf : ∀ x,|f x|≤H) (hg : ∀ y,|g y|≤C) : |
| |(∑ e,G.fiberWeight e*∑ y,prob (G.targets e) y*(f (G.source e)*g y))- |
| (∑ x,prob Finset.univ x*f x)*(∑ y,G.incomingWeight y*g y)|≤ |
| C*(testDiscrep (G.retainedWeight T) (G.retainedSet T) f+2*H*G.omittedMass T) := by |
| have hC : 0≤C := (abs_nonneg (g (Classical.choice (Fintype.card_pos_iff.mp (by omega))))).trans (hg _) |
| rw [← G.incoming_eval (fun x y => f x*g y)] |
| have he : (∑ y,G.incomingWeight y*∑ x,G.conditionalSource y x*(f x*g y))- |
| (∑ x,prob Finset.univ x*f x)*(∑ y,G.incomingWeight y*g y)= |
| ∑ y,G.incomingWeight y*((∑ x,G.conditionalSource y x*f x)-∑ x,prob Finset.univ x*f x)*g y := by |
| simp only [Finset.mul_sum,Finset.sum_mul,← Finset.sum_sub_distrib] |
| apply Finset.sum_congr rfl |
| intro y _ |
| apply Finset.sum_congr rfl |
| intro x _ |
| ring |
| rw [he] |
| calc |
| _ ≤ ∑ y,|G.incomingWeight y*((∑ x,G.conditionalSource y x*f x)-∑ x,prob Finset.univ x*f x)*g y| := |
| Finset.abs_sum_le_sum_abs _ _ |
| _ ≤ C*(∑ y,G.incomingWeight y*|(∑ x,G.conditionalSource y x*f x)-∑ x,prob Finset.univ x*f x|) := by |
| rw [Finset.mul_sum] |
| apply Finset.sum_le_sum |
| intro y _ |
| rw [abs_mul,abs_mul,abs_of_nonneg (G.incomingWeight_nonneg y)] |
| nlinarith [mul_le_mul_of_nonneg_left (hg y) (mul_nonneg (G.incomingWeight_nonneg y) (abs_nonneg ((∑ x,G.conditionalSource y x*f x)-∑ x,prob Finset.univ x*f x)))] |
| _ ≤ _ := mul_le_mul_of_nonneg_left (incoming_discrep_bound G T H hT hn f hf) hC |
|
|
| end WeakPinned.Bounded |
|
|
| end |
| end |
| section |
| noncomputable section |
| open MeasureTheory Filter |
| open scoped BigOperators Topology BoundedContinuousFunction |
| namespace WeakPinned.Bounded |
| open Variance |
| local instance : MeasurableSpace Sphere := borel Sphere |
| local instance : BorelSpace Sphere := ⟨rfl⟩ |
| local instance : MeasurableSingletonClass (Quad Sphere) := inferInstance |
| local instance : MeasurableSingletonClass ((Sphere × Sphere) × (Sphere × Sphere)) := inferInstance |
| local instance : MeasurableSingletonClass (Expanded Sphere) := Prod.instMeasurableSingletonClass |
| variable {V : Type u8} {I : Type u9} [Fintype V] [DecidableEq V] [Fintype I] |
| variable (G : Core.Fibers V I) |
|
|
| omit G [Fintype I] in |
| lemma integral_uniformLaw (U : V → Sphere) (f : Sphere → ℝ) : |
| (∫ z,f z ∂ uniformLaw U)=∑ x,prob Finset.univ x*f (U x) := by |
| rw [uniformLaw,integral_finiteLaw _ _ (fun _ => inv_nonneg.mpr (Nat.cast_nonneg _))] |
| simp only [prob,Finset.mem_univ,ite_true,Finset.card_univ,one_div] |
|
|
| lemma integral_edgeProbability (U₁ U₂ : V → Sphere) (hG : 0<G.edgeCount) |
| (hB : ∀ e,(G.targets e).Nonempty) (f : Quad Sphere → ℝ) : |
| (∫ q,f q ∂ (edgeProbability G U₁ U₂ hG hB:Measure (Quad Sphere)))= |
| ∑ e,G.fiberWeight e*∑ y,prob (G.targets e) y*f (edgeValue G.source U₁ U₂ (e,y)) := by |
| change (∫ q,f q ∂ finiteLaw _ _)=_ |
| rw [integral_finiteLaw _ _ (graph_edgeWeight_nonneg G)] |
| simp only [edgeWeight,Fintype.sum_prod_type,mul_assoc,← Finset.mul_sum] |
|
|
| lemma integral_expandedProbability (U₁ U₂ : V → Sphere) (hG : 0<G.edgeCount) |
| (hB : ∀ e,(G.targets e).Nonempty) (f : Expanded Sphere → ℝ) : |
| (∫ r,f r ∂ (expandedProbability G U₁ U₂ hG hB:Measure (Expanded Sphere)))= |
| ∑ r,tripleWeight G.fiberWeight (fun e => prob (G.targets e)) r*f (tripleValue G.source U₁ U₂ r) := by |
| change (∫ r,f r ∂ finiteLaw (tripleWeight G.fiberWeight (fun e => prob (G.targets e))) (tripleValue G.source U₁ U₂))=_ |
| exact integral_finiteLaw _ _ (graph_tripleWeight_nonneg G) f |
|
|
| lemma expanded_test_bound (U₁ U₂ : V → Sphere) (hG : 0<G.edgeCount) |
| (hB : ∀ e,(G.targets e).Nonempty) (hn : 2≤Fintype.card V) |
| (π : Sphere × Sphere → Sphere) (f : Quad Sphere → ℝ) (g h : Sphere → ℝ) (C H₁ H₂ : ℝ) |
| (hC : 0≤C) (hf : ∀ q,|f q|≤C) (hg : ∀ z,|g z|≤H₁) (hh : ∀ z,|h z|≤H₂) : |
| |(∫ r,f r.1*g (π r.2.1)*h (π r.2.2) ∂ (expandedProbability G U₁ U₂ hG hB:Measure (Expanded Sphere)))- |
| (∫ q,f q ∂ (edgeProbability G U₁ U₂ hG hB:Measure (Quad Sphere)))* |
| (∫ z,g z ∂ uniformLaw (fun x => π (U₁ x,U₂ x)))*(∫ z,h z ∂ uniformLaw (fun x => π (U₁ x,U₂ x)))|≤ |
| C*(H₂*testDiscrep G.fiberWeight G.targets (fun x => g (π (U₁ x,U₂ x)))+ |
| H₁*testDiscrep G.fiberWeight G.targets (fun x => h (π (U₁ x,U₂ x)))) := by |
| rw [integral_expandedProbability,integral_edgeProbability,integral_uniformLaw,integral_uniformLaw] |
| have hA : |∑ x,prob (Finset.univ : Finset V) x*g (π (U₁ x,U₂ x))|≤H₁ := by |
| exact avg_abs_bound _ _ _ (Core.Fibers.prob_nn _) (sum_prob _ |
| (Finset.univ_nonempty_iff.mpr (Fintype.card_pos_iff.mp (by omega)))) (fun x => hg _) |
| have hb := mixture_pair_test_bound G.fiberWeight (fun e => prob (G.targets e)) |
| (fun e y => f (edgeValue G.source U₁ U₂ (e,y))) (fun x => g (π (U₁ x,U₂ x))) (fun x => h (π (U₁ x,U₂ x))) |
| (∑ x,prob Finset.univ x*g (π (U₁ x,U₂ x))) (∑ x,prob Finset.univ x*h (π (U₁ x,U₂ x))) |
| C H₁ H₂ G.fiberWeight_nonneg (fun e => Core.Fibers.prob_nn _) (fun e => sum_prob _ (hB e)) |
| hC (fun e y => hf _) (fun x => hh _) hA |
| simpa only [tripleValue,testDiscrep,edgeWeight,Fintype.sum_prod_type,mul_assoc,← Finset.mul_sum] using hb |
|
|
| lemma integral_target (U₁ U₂ : V → Sphere) (hG : 0<G.edgeCount) |
| (hB : ∀ e,(G.targets e).Nonempty) (g : Sphere × Sphere → ℝ) : |
| (∫ q,g q.2.2 ∂ (edgeProbability G U₁ U₂ hG hB:Measure (Quad Sphere)))= |
| ∑ y,G.incomingWeight y*g (U₁ y,U₂ y) := by |
| rw [integral_edgeProbability] |
| simp only [edgeValue,Finset.mul_sum,← mul_assoc] |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro y _ |
| rw [← Finset.sum_mul,G.fiber_marginal,G.incoming_marginal] |
|
|
| lemma incoming_law_test_bound (U₁ U₂ : V → Sphere) (hG : 0<G.edgeCount) |
| (hB : ∀ e,(G.targets e).Nonempty) (hn : 2≤Fintype.card V) (T H C : ℝ) (hT : 0<T) |
| (f : Sphere → ℝ) (g : Sphere × Sphere → ℝ) (hf : ∀ z,|f z|≤H) (hg : ∀ z,|g z|≤C) : |
| |(∫ q,f q.1*g q.2.2 ∂ (edgeProbability G U₁ U₂ hG hB:Measure (Quad Sphere)))- |
| (∫ z,f z ∂ uniformLaw U₁)*(∫ q,g q.2.2 ∂ (edgeProbability G U₁ U₂ hG hB:Measure (Quad Sphere)))|≤ |
| C*(testDiscrep (G.retainedWeight T) (G.retainedSet T) (fun x => f (U₁ x))+2*H*G.omittedMass T) := by |
| rw [integral_target,integral_edgeProbability,integral_uniformLaw] |
| exact incoming_test_bound G T H C hT hn (fun x => f (U₁ x)) (fun x => g (U₁ x,U₂ x)) |
| (fun x => hf _) (fun x => hg _) |
|
|
| end WeakPinned.Bounded |
|
|
| end |
| end |
| section |
| noncomputable section |
| open MeasureTheory Filter |
| open scoped BigOperators Topology BoundedContinuousFunction |
| namespace WeakPinned.Bounded |
| local instance : MeasurableSpace Sphere := borel Sphere |
| local instance : BorelSpace Sphere := ⟨rfl⟩ |
| variable {V : ℕ → Type u10} {I : ℕ → Type u11} [∀ n,Fintype (V n)] [∀ n,DecidableEq (V n)] [∀ n,Fintype (I n)] |
| variable (G : ∀ n,Core.Fibers (V n) (I n)) (U₁ U₂ : ∀ n,V n → Sphere) |
| (hG : ∀ n,0<(G n).edgeCount) (hB : ∀ n e,((G n).targets e).Nonempty) |
| (hn : ∀ n,2≤Fintype.card (V n)) |
|
|
| include hn |
| lemma expanded_test_rate (π : Sphere × Sphere → Sphere) |
| (hd : ∀ f : C(Sphere,ℝ),Tendsto (fun n => testDiscrep (G n).fiberWeight (G n).targets |
| (fun x => f (π (U₁ n x,U₂ n x)))) atTop (𝓝 0)) |
| (f : Quad Sphere →ᵇ ℝ) (g h : Sphere →ᵇ ℝ) : |
| Tendsto (fun n => (∫ r,f r.1*g (π r.2.1)*h (π r.2.2) |
| ∂ (expandedProbability (G n) (U₁ n) (U₂ n) (hG n) (hB n):Measure (Expanded Sphere)))- |
| (∫ q,f q ∂ (edgeProbability (G n) (U₁ n) (U₂ n) (hG n) (hB n):Measure (Quad Sphere)))* |
| (∫ z,g z ∂ uniformLaw (fun x => π (U₁ n x,U₂ n x)))* |
| (∫ z,h z ∂ uniformLaw (fun x => π (U₁ n x,U₂ n x)))) atTop (𝓝 0) := by |
| apply (tendsto_zero_iff_abs_tendsto_zero _).mpr |
| have ht := ((hd g.toContinuousMap).const_mul ‖h‖ |>.add |
| ((hd h.toContinuousMap).const_mul ‖g‖)).const_mul ‖f‖ |
| simp only [mul_zero,zero_add] at ht |
| apply squeeze_zero (fun n => abs_nonneg _) (fun n => ?_) ht |
| exact expanded_test_bound (G n) (U₁ n) (U₂ n) (hG n) (hB n) (hn n) π f g h |
| ‖f‖ ‖g‖ ‖h‖ (norm_nonneg _) (fun q => by simpa only [Real.norm_eq_abs] using f.norm_coe_le_norm q) |
| (fun z => by simpa only [Real.norm_eq_abs] using g.norm_coe_le_norm z) |
| (fun z => by simpa only [Real.norm_eq_abs] using h.norm_coe_le_norm z) |
|
|
| lemma incoming_law_test_rate (T : ℕ → ℝ) (hT : ∀ n,0<T n) |
| (hd : ∀ f : C(Sphere,ℝ),Tendsto (fun n => testDiscrep ((G n).retainedWeight (T n)) |
| ((G n).retainedSet (T n)) (fun x => f (U₁ n x))) atTop (𝓝 0)) |
| (ho : Tendsto (fun n => (G n).omittedMass (T n)) atTop (𝓝 0)) |
| (f : Sphere →ᵇ ℝ) (g : Sphere × Sphere →ᵇ ℝ) : |
| Tendsto (fun n => (∫ q,f q.1*g q.2.2 ∂ (edgeProbability (G n) (U₁ n) (U₂ n) (hG n) (hB n):Measure (Quad Sphere)))- |
| (∫ z,f z ∂ uniformLaw (U₁ n))*(∫ q,g q.2.2 ∂ (edgeProbability (G n) (U₁ n) (U₂ n) (hG n) (hB n):Measure (Quad Sphere)))) |
| atTop (𝓝 0) := by |
| apply (tendsto_zero_iff_abs_tendsto_zero _).mpr |
| have ht := ((hd f.toContinuousMap).add (ho.const_mul (2*‖f‖))).const_mul ‖g‖ |
| simp only [mul_zero,zero_add] at ht |
| apply squeeze_zero (fun n => abs_nonneg _) (fun n => ?_) ht |
| exact incoming_law_test_bound (G n) (U₁ n) (U₂ n) (hG n) (hB n) (hn n) (T n) ‖f‖ ‖g‖ (hT n) f g |
| (fun z => by simpa only [Real.norm_eq_abs] using f.norm_coe_le_norm z) |
| (fun z => by simpa only [Real.norm_eq_abs] using g.norm_coe_le_norm z) |
|
|
| end WeakPinned.Bounded |
|
|
| end |
| end |
|
|
| end OAI |
|
|
