| import Mathlib |
| import OAI.Probability.SKGap.Localization.ConditionalModelAlgebra |
|
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| namespace OAI |
|
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| section |
| noncomputable section |
| namespace SKGap |
| open Matrix Real Set MeasureTheory ProbabilityTheory |
| open scoped BigOperators Matrix.Norms.Frobenius |
|
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| def conditionalMiddleSize (j V W : ℝ) (p : ScalarPoint) : ℝ := |
| (1+2*|conditionalKappa j p-1|+|conditionalEll j p-2*conditionalKappa j p+1|)*W+ |
| 2*(|j*sqrt (scalarQMoment p.1/scalarS j p)| * (sqrt (scalarS j p)*V)+ |
| |j*scalarA j p/scalarS j p|)+ |
| |2*j*(scalarA j p+j*scalarBMoment p.1*scalarQMoment p.1)/(scalarS j p+j*scalarQMoment p.1)|+ |
| |j| * V^2+|2*j*scalarQMoment p.1|+|j*scalarBMoment p.1| |
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| lemma continuousAt_conditionalMiddleSize {X : Type*} [TopologicalSpace X] |
| {R : ℝ} (hR : 0 ≤ R) (j V W : ℝ) (f : X → ScalarPoint) (hf : Continuous f) |
| (hdom : ∀ x,f x ∈ scalarMomentDomain R) {x : X} |
| (hs : scalarS j (f x) ≠ 0) (hS : scalarS j (f x)+j*scalarQMoment (f x).1 ≠ 0) : |
| ContinuousAt (fun x=>conditionalMiddleSize j V W (f x)) x := by |
| have hq := (continuous_scalarQMoment.comp (continuous_fst.comp hf)).continuousAt (x := x) |
| have hb := (continuous_scalarBMoment.comp (continuous_fst.comp hf)).continuousAt (x := x) |
| have hss := ((continuous_scalarS j).comp hf).continuousAt (x := x) |
| have hSS : ContinuousAt (fun x=>scalarS j (f x)+j*scalarQMoment (f x).1) x := hss.add (continuousAt_const.mul hq) |
| have ha := ((continuousOn_scalarA hR j).comp_continuous hf hdom).continuousAt (x := x) |
| have hn : ContinuousAt (fun x=>(f x).2.1+(f x).2.2^2) x := by fun_prop |
| have hk : ContinuousAt (fun x=>conditionalKappa j (f x)) x := (hn.div hss hs).sqrt |
| have hl : ContinuousAt (fun x=>conditionalEll j (f x)) x := (hn.div hSS hS).sqrt |
| unfold conditionalMiddleSize |
| fun_prop (disch := assumption) |
|
|
| theorem conditionalMiddleSize_uniform {j R t0 T : ℝ} (hj : 0 ≤ j) (hR : 0 ≤ R) |
| (ht0 : 0 < t0) (V W : ℝ) : |
| ∃ L : ℝ, 1 ≤ L ∧ ∀ (P : ProbabilityMeasure ℝ), P ∈ momentBall R → |
| ∀ t ∈ Icc t0 T,∀ σ : ℝ, |σ| ≤ 1 → conditionalMiddleSize j V W (P,t,σ) ≤ L := by |
| let X : Type := ↥(momentBall R) × ↥(Icc t0 T) × ↥(Icc (-1:ℝ) 1) |
| let : CompactSpace (momentBall R) := isCompact_iff_compactSpace.mp (isCompact_momentBall R) |
| let f : X → ScalarPoint := fun x=>((x.1:ProbabilityMeasure ℝ),(x.2.1:ℝ),(x.2.2:ℝ)) |
| have hf : Continuous f := by dsimp [f,X]; fun_prop |
| have hdom : ∀ x,f x ∈ scalarMomentDomain R := fun (x : X)=>⟨x.1.2,mem_univ _⟩ |
| have hc : Continuous (fun x : X=>conditionalMiddleSize j V W (f x)) := by |
| apply continuous_iff_continuousAt.mpr |
| intro x |
| have ht : 0 < (x.2.1:ℝ) := ht0.trans_le x.2.1.2.1 |
| have hs : 0 < scalarS j (f x) := by |
| dsimp [scalarS,f] |
| exact add_pos_of_pos_of_nonneg (add_pos_of_pos_of_nonneg ht (sq_nonneg _)) |
| (mul_nonneg hj (scalarQMoment_bounds _).1) |
| have hS : 0 < scalarS j (f x)+j*scalarQMoment (f x).1 := |
| add_pos_of_pos_of_nonneg hs (mul_nonneg hj (scalarQMoment_bounds _).1) |
| exact continuousAt_conditionalMiddleSize hR j V W f hf hdom hs.ne' hS.ne' |
| obtain ⟨M,hM⟩ := isCompact_univ.bddAbove_image hc.continuousOn |
| refine ⟨max 1 M,le_max_left _ _,?_⟩ |
| intro P hP t ht σ hσ |
| let x : X := (⟨P,hP⟩,⟨t,ht⟩,⟨σ,abs_le.mp hσ⟩) |
| exact (hM (mem_image_of_mem _ (mem_univ x))).trans (le_max_right _ _) |
|
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| lemma empirical_column_coefficient {n : ℕ} [NeZero n] {j t0 t T σ : ℝ} |
| (hj : 0 ≤ j) (ht0 : 0 < t0) (ht : t ∈ Icc t0 T) (y : Fin n → ℝ) : |
| (1+|scalarD j (empiricalLaw y,t,σ)|)/scalarS j (empiricalLaw y,t,σ) ≤ |
| max 1 ((1+|T|+j)/t0) := by |
| have ht' : 0 < t := ht0.trans_le ht.1 |
| have hd : |scalarD j (empiricalLaw y,t,σ)| ≤ |T|+j := by |
| apply (abs_add_le _ _).trans |
| rw [abs_of_pos ht',abs_mul,abs_of_nonneg hj] |
| have hh := mul_le_mul_of_nonneg_left (scalarM_bounds (empiricalLaw y)) hj |
| have hT := ht.2.trans (le_abs_self T) |
| nlinarith only [hh,hT] |
| have hs : t0 ≤ scalarS j (empiricalLaw y,t,σ) := by |
| dsimp [scalarS] |
| exact ht.1.trans (le_add_of_nonneg_right (sq_nonneg σ) |>.trans |
| (le_add_of_nonneg_right (mul_nonneg hj (scalarQMoment_bounds _).1))) |
| apply le_trans _ (le_max_right _ _) |
| exact div_le_div₀ (by positivity) (by linarith) ht0 hs |
| end SKGap |
| end |
| end |
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| end OAI |
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