| import Mathlib |
| import OAI.Probability.SKGap.Localization.ConditionalModelAlgebra |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Set Filter Topology |
|
|
| theorem compact_perturbation_tube {X : Type*} [TopologicalSpace X] |
| {K : Set X} (hK : IsCompact K) (E : Set (X × ℝ)) |
| (hE : ∀ x ∈ K, E ∈ 𝓝 (x,0)) : |
| ∃ U : Set X, IsOpen U ∧ K ⊆ U ∧ ∃ δ > 0, |
| ∀ x ∈ U, ∀ r : ℝ, |r| ≤ δ → (x,r) ∈ E := by |
| have hc : K ×ˢ ({0} : Set ℝ) ⊆ interior E := by |
| rintro ⟨x,r⟩ ⟨hx,hr⟩ |
| have hr0 : r=0 := mem_singleton_iff.mp hr |
| subst r |
| exact mem_interior_iff_mem_nhds.mpr (hE x hx) |
| obtain ⟨U,V,hU,hV,hKU,h0V,hUV⟩ := generalized_tube_lemma hK isCompact_singleton isOpen_interior hc |
| have hV0 : V ∈ 𝓝 (0:ℝ) := hV.mem_nhds (h0V (mem_singleton 0)) |
| obtain ⟨r,hr,hrV⟩ := Metric.mem_nhds_iff.mp hV0 |
| refine ⟨U,hU,hKU,r/2,half_pos hr,?_⟩ |
| intro x hx a ha |
| apply interior_subset (hUV ⟨hx,?_⟩) |
| apply hrV |
| rw [Metric.mem_ball,Real.dist_eq,sub_zero] |
| linarith |
| end SKGap |
| namespace SKGap |
| open Set Filter Topology |
| lemma compact_perturbation_tube_norm {X Y : Type*} [TopologicalSpace X] |
| [NormedAddCommGroup Y] {K : Set X} (hK : IsCompact K) (E : Set (X × Y)) |
| (hE : ∀ x ∈ K, E ∈ 𝓝 (x,0)) : |
| ∃ U : Set X, IsOpen U ∧ K ⊆ U ∧ ∃ δ > 0, |
| ∀ x ∈ U, ∀ r : Y, ‖r‖ ≤ δ → (x,r) ∈ E := by |
| have hc : K ×ˢ ({0} : Set Y) ⊆ interior E := by |
| rintro ⟨x,r⟩ ⟨hx,hr⟩ |
| have hr0 : r=0 := mem_singleton_iff.mp hr |
| subst r |
| exact mem_interior_iff_mem_nhds.mpr (hE x hx) |
| obtain ⟨U,V,hU,hV,hKU,h0V,hUV⟩ := generalized_tube_lemma hK isCompact_singleton isOpen_interior hc |
| obtain ⟨r,hr,hrV⟩ := Metric.mem_nhds_iff.mp (hV.mem_nhds (h0V (mem_singleton 0))) |
| refine ⟨U,hU,hKU,r/2,half_pos hr,?_⟩ |
| intro x hx a ha |
| apply interior_subset (hUV ⟨hx,hrV ?_⟩) |
| rw [Metric.mem_ball,dist_zero_right] |
| linarith |
| end SKGap |
| end |
| end |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Real Matrix MeasureTheory ProbabilityTheory Set Filter Topology |
| open scoped BigOperators Matrix.Norms.Frobenius |
|
|
| def conditionalScalarSize (j : ℝ) (p : ScalarPoint) (r : ℝ) : ℝ := |
| (1+|scalarD j p|)/scalarS j p+1/scalarQMoment p.1+1/scalarS j p+ |
| 1/(j*scalarBMoment p.1+r)+(j*scalarBMoment p.1+r)+‖conditionalScalarData j p‖ |
|
|
| lemma continuousAt_conditionalSize {X : Type*} [TopologicalSpace X] |
| {R j : ℝ} (hR : 0 ≤ R) (f : X → ScalarPoint) (hf : Continuous f) |
| (hdom : ∀ x,f x ∈ scalarMomentDomain R) (r : X → ℝ) (hr : Continuous r) |
| {x : X} (hq : 0 < scalarQMoment (f x).1) (hs : 0 < scalarS j (f x)) |
| (hS : 0 < scalarS j (f x)+j*scalarQMoment (f x).1) |
| (hB : 0 < j*scalarBMoment (f x).1+r x) : |
| ContinuousAt (fun x=>conditionalScalarSize j (f x) (r x)) x := by |
| have hq' := (continuous_scalarQMoment.comp (continuous_fst.comp hf)).continuousAt (x := x) |
| have hs' := ((continuous_scalarS j).comp hf).continuousAt (x := x) |
| have hd' := ((continuous_scalarD j).comp hf).continuousAt (x := x) |
| have hb' := (((continuous_const : Continuous (fun _ : X=>j)).mul (continuous_scalarBMoment.comp (continuous_fst.comp hf))).add hr).continuousAt (x := x) |
| have he' := (continuousAt_conditionalData hR f hf hdom r hr hq hs hS).2.norm |
| exact (((((continuousAt_const.add hd'.abs).div hs' hs.ne').add |
| (continuousAt_const.div hq' hq.ne')).add (continuousAt_const.div hs' hs.ne')).add |
| (continuousAt_const.div hb' hB.ne')).add hb' |>.add he' |
|
|
| lemma continuousAt_augmentedGram {X : Type*} [TopologicalSpace X] |
| {H : X → Matrix (Fin 3) (Fin 3) ℝ} {x : X} (hH : ContinuousAt H x) : |
| ContinuousAt (fun x=>augmentedGram (ν := Unit) (H x)) x := by |
| apply continuousAt_pi.mpr |
| intro i |
| apply continuousAt_pi.mpr |
| intro k |
| cases i with |
| | inl i => cases k with |
| | inl k => exact (continuousAt_pi.mp (continuousAt_pi.mp hH i) k) |
| | inr k => exact continuousAt_const |
| | inr i => cases k <;> exact continuousAt_const |
|
|
| abbrev ScalarTimeDomain (R t0 T : ℝ) := (momentBall R) × (Icc t0 T) |
|
|
| def scalarEqualitySet (j R t0 T : ℝ) : Set (ScalarTimeDomain R t0 T) := |
| {x | (x.1 : ProbabilityMeasure ℝ)=scalarCurve j x.2} |
|
|
| def scalarPerturbedPoint {R t0 T : ℝ} (z : ScalarTimeDomain R t0 T × (ℝ × ℝ)) : ScalarPoint := |
| ((z.1.1 : ProbabilityMeasure ℝ),(z.1.2 : ℝ),z.2.1) |
|
|
| lemma continuous_scalarPerturbedPoint {R t0 T : ℝ} : |
| Continuous (@scalarPerturbedPoint R t0 T) := by |
| unfold scalarPerturbedPoint |
| fun_prop |
|
|
| lemma scalarPerturbedPoint_domain {R t0 T : ℝ} (z : ScalarTimeDomain R t0 T × (ℝ × ℝ)) : |
| scalarPerturbedPoint z ∈ scalarMomentDomain R := ⟨z.1.1.2,mem_univ _⟩ |
|
|
| lemma isCompact_scalarEqualitySet {j R t0 T : ℝ} (hj : 0 < j) (hj1 : j < 1) (ht0 : 0 < t0) : |
| IsCompact (scalarEqualitySet j R t0 T) := by |
| let : CompactSpace (momentBall R) := isCompact_iff_compactSpace.mp (isCompact_momentBall R) |
| have hcurve : Continuous (fun t : Icc t0 T=>scalarCurve j t) := |
| (continuous_scalarCurve_nonneg hj hj1).comp (continuous_subtype_val.subtype_mk (fun t=>ht0.le.trans t.2.1)) |
| exact (isClosed_eq (continuous_subtype_val.comp continuous_fst) (hcurve.comp continuous_snd)).isCompact |
|
|
| theorem scalar_conditional_neighborhood {j R t0 T ρ : ℝ} |
| (hj : 0 < j) (hj1 : j < 1) (hR : 0 ≤ R) (ht0 : 0 < t0) (hρ : 0 < ρ) : |
| ∃ L : ℝ, 1 ≤ L ∧ ∃ U : Set (ScalarTimeDomain R t0 T), IsOpen U ∧ |
| scalarEqualitySet j R t0 T ⊆ U ∧ ∃ δ > 0, |
| ∀ x ∈ U,∀ σ r : ℝ, |σ| ≤ δ → |r| ≤ δ → |
| let p : ScalarPoint := ((x.1 : ProbabilityMeasure ℝ),(x.2 : ℝ),σ) |
| 0 < scalarQMoment p.1 ∧ 0 < scalarS j p ∧ 0 < j*scalarBMoment p.1+r ∧ |
| conditionalScalarSize j p r < L ∧ |
| ‖augmentedGram (ν := Unit) (scalarMomentGram p.1 (scalarD j p) (scalarS j p))- |
| augmentedGram (ν := Unit) (limitingGram (scalarBMoment (scalarCurve j x.2)) |
| (scalarEMoment (scalarCurve j x.2)) (scalarFixedPoint j x.2))‖ < ρ ∧ |
| ‖conditionalMeanData j p r-curveCoefficient j x.2‖ < ρ := by |
| let X := ScalarTimeDomain R t0 T |
| let K := scalarEqualitySet j R t0 T |
| let f : X × (ℝ × ℝ) → ScalarPoint := scalarPerturbedPoint |
| let r : X × (ℝ × ℝ) → ℝ := fun z=>z.2.2 |
| have hf : Continuous f := continuous_scalarPerturbedPoint |
| have hr : Continuous r := continuous_snd.comp continuous_snd |
| have hd : ∀ z,f z ∈ scalarMomentDomain R := scalarPerturbedPoint_domain |
| have hK : IsCompact K := isCompact_scalarEqualitySet hj hj1 ht0 |
| have heq (x : X) (hx : x ∈ K) : f (x,0)=(scalarCurve j x.2,(x.2:ℝ),0) := by |
| change ((x.1:ProbabilityMeasure ℝ),(x.2:ℝ),0)=_ |
| rw [show (x.1:ProbabilityMeasure ℝ)=scalarCurve j x.2 from hx] |
| have hpoint (x : X) (hx : x ∈ K) : |
| 0 < scalarQMoment (f (x,0)).1 ∧ 0 < scalarS j (f (x,0)) ∧ |
| 0 < scalarS j (f (x,0))+j*scalarQMoment (f (x,0)).1 ∧ |
| 0 < j*scalarBMoment (f (x,0)).1+r (x,0) := by |
| rw [heq x hx,scalarQMoment_curve hj hj1 (ht0.le.trans x.2.2.1),scalarS_curve hj hj1 (ht0.le.trans x.2.2.1)] |
| have hp := scalar_fixed_pos hj hj1 (ht0.trans_le x.2.2.1) |
| exact ⟨hp.2,hp.1,add_pos hp.1 (mul_pos hj hp.2),by simpa [r] using mul_pos hj (scalarBMoment_pos (scalarCurve j x.2))⟩ |
| have hsize (x : X) (hx : x ∈ K) : |
| ContinuousAt (fun z=>conditionalScalarSize j (f z) (r z)) (x,0) := |
| continuousAt_conditionalSize hR f hf hd r hr (hpoint x hx).1 (hpoint x hx).2.1 |
| (hpoint x hx).2.2.1 (hpoint x hx).2.2.2 |
| have hsizeK : ContinuousOn (fun x : X=>conditionalScalarSize j (f (x,0)) (r (x,0))) K := by |
| intro x hx |
| exact ((hsize x hx).comp (x := x) (continuousAt_id.prodMk continuousAt_const)).continuousWithinAt |
| obtain ⟨M,hM⟩ := hK.bddAbove_image hsizeK |
| let L := max 1 (M+1) |
| have hL : 1 ≤ L := le_max_left _ _ |
| have hsmall (x : X) (hx : x ∈ K) : conditionalScalarSize j (f (x,0)) (r (x,0)) < L := |
| (lt_of_le_of_lt (hM (mem_image_of_mem _ hx)) (by linarith : M < M+1)).trans_le (le_max_right _ _) |
| obtain ⟨hs,hq,hb,he,hkap,hc⟩ := continuous_curve_statistics (T := T) hj hj1 ht0 |
| let G : X × (ℝ × ℝ) → Matrix ((Fin 3) ⊕ Unit) ((Fin 3) ⊕ Unit) ℝ := fun z=> |
| augmentedGram (ν := Unit) (scalarMomentGram (f z).1 (scalarD j (f z)) (scalarS j (f z))) |
| let G0 : X × (ℝ × ℝ) → Matrix ((Fin 3) ⊕ Unit) ((Fin 3) ⊕ Unit) ℝ := fun z=> |
| augmentedGram (ν := Unit) (limitingGram (scalarBMoment (scalarCurve j z.1.2)) |
| (scalarEMoment (scalarCurve j z.1.2)) (scalarFixedPoint j z.1.2)) |
| have hG0 : Continuous G0 := by |
| apply continuous_iff_continuousAt.mpr |
| intro z |
| apply continuousAt_augmentedGram |
| unfold limitingGram |
| apply continuousAt_pi.mpr |
| intro i |
| apply continuousAt_pi.mpr |
| intro k |
| have hb' := (hb.comp (continuous_snd.comp continuous_fst)).continuousAt (x := z) |
| have he' := (he.comp (continuous_snd.comp continuous_fst)).continuousAt (x := z) |
| have hs' := (hs.comp (continuous_snd.comp continuous_fst)).continuousAt (x := z) |
| fin_cases i <;> fin_cases k <;> simp only [Matrix.of_apply,Matrix.cons_val,Fin.reduceFinMk] <;> |
| fun_prop (disch := exact (scalar_fixed_pos hj hj1 (ht0.trans_le z.1.2.2.1)).1.ne') |
| let E : Set (X × (ℝ × ℝ)) := {z | |
| 0 < scalarQMoment (f z).1 ∧ 0 < scalarS j (f z) ∧ 0 < j*scalarBMoment (f z).1+r z ∧ |
| conditionalScalarSize j (f z) (r z) < L ∧ ‖G z-G0 z‖ < ρ ∧ |
| ‖conditionalMeanData j (f z) (r z)-curveCoefficient j z.1.2‖ < ρ} |
| have hE (x : X) (hx : x ∈ K) : E ∈ 𝓝 (x,0) := by |
| have hpt := hpoint x hx |
| have hqc := (continuous_scalarQMoment.comp (continuous_fst.comp hf)).continuousAt (x := (x,0)) |
| have hsc := ((continuous_scalarS j).comp hf).continuousAt (x := (x,0)) |
| have hbc := (((continuous_const : Continuous (fun _ : X × (ℝ × ℝ)=>j)).mul (continuous_scalarBMoment.comp (continuous_fst.comp hf))).add hr).continuousAt (x := (x,0)) |
| have cg : ContinuousAt G (x,0) := by |
| apply continuousAt_augmentedGram |
| exact (continuousAt_scalarMomentGram (p := ((f (x,0)).1,scalarD j (f (x,0)),scalarS j (f (x,0)))) hpt.2.1.ne').comp |
| (x := (x,0)) ((continuous_fst.comp hf).continuousAt.prodMk |
| (((continuous_scalarD j).comp hf).continuousAt.prodMk hsc)) |
| have cm := (continuousAt_conditionalData hR f hf hd r hr hpt.1 hpt.2.1 hpt.2.2.1).1 |
| have hgzero : G (x,0)-G0 (x,0)=0 := by |
| dsimp only [G,G0] |
| rw [heq x hx,scalarD_curve hj hj1 (ht0.le.trans x.2.2.1),scalarS_curve hj hj1 (ht0.le.trans x.2.2.1), |
| scalarMomentGram_curve hj hj1 (ht0.trans_le x.2.2.1)] |
| exact sub_self _ |
| have hczero : conditionalMeanData j (f (x,0)) (r (x,0))-curveCoefficient j x.2=0 := by |
| rw [heq x hx] |
| change conditionalMeanData j (scalarCurve j x.2,(x.2:ℝ),0) 0-curveCoefficient j x.2=0 |
| rw [conditionalMeanData_curve hj hj1 (ht0.trans_le x.2.2.1),sub_self] |
| have e1 := hqc.eventually_const_lt hpt.1 |
| have e2 := hsc.eventually_const_lt hpt.2.1 |
| have e3 := hbc.eventually_const_lt hpt.2.2.2 |
| have e4 := (hsize x hx).eventually_lt_const (hsmall x hx) |
| have e5 := (cg.sub hG0.continuousAt).norm.eventually_lt_const (by change ‖G (x,0)-G0 (x,0)‖ < ρ; simpa only [hgzero,norm_zero] using hρ) |
| have e6 := (cm.sub (hc.comp (continuous_snd.comp continuous_fst)).continuousAt).norm.eventually_lt_const (by change ‖conditionalMeanData j (f (x,0)) (r (x,0))-curveCoefficient j x.2‖ < ρ; simpa only [hczero,norm_zero] using hρ) |
| exact e1.and (e2.and (e3.and (e4.and (e5.and e6)))) |
| obtain ⟨U,hU,hKU,δ,hδ,hδE⟩ := compact_perturbation_tube_norm hK E hE |
| refine ⟨L,hL,U,hU,hKU,δ,hδ,?_⟩ |
| intro x hx σ r hσ hr |
| exact hδE x hx (σ,r) (by simpa only [Prod.norm_def,Real.norm_eq_abs,max_le_iff] using And.intro hσ hr) |
| end SKGap |
| end |
| end |
|
|
| end OAI |
|
|
