| import Mathlib |
| import OAI.Probability.SKGap.Gaussian.GaussianCompactTailRate |
| import OAI.Probability.SKGap.Localization.ScalarRelativeIsolation |
| import OAI.Probability.SKGap.Stability.ConditionalStability |
| import OAI.Probability.SKGap.Localization.ConditionalWeight |
| import OAI.Probability.SKGap.Entropy.WeightedRateAssembly |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Matrix Real Set MeasureTheory ProbabilityTheory GaussianDensity |
| open scoped BigOperators Matrix.Norms.Frobenius ENNReal |
|
|
| def badFieldFlat (n : ℕ) (j A ε m : ℝ) : Set (((Fin n×Fin n) → ℝ) × (Fin n → ℝ)) := |
| {p | ((fun i k=>p.1 (i,k)),p.2)∈badFieldSet n j A ε m} |
|
|
| lemma measurableSet_badFieldFlat (n : ℕ) (j A ε m : ℝ) : |
| MeasurableSet (badFieldFlat n j A ε m) := |
| ((isClosed_badFieldSet n j A ε m).preimage (by fun_prop)).measurableSet |
|
|
| def reducedBadDensity {n : ℕ} [NeZero n] (j t σ A ε m : ℝ) (y : Fin n → ℝ) : ℝ≥0∞ := |
| ENNReal.ofReal (sqrt (scalarS j (scalarEmpirical y t σ)/ |
| (scalarS j (scalarEmpirical y t σ)+j*scalarQMoment (empiricalLaw y)))*scalarIntegrand j t σ y)* |
| conditionalFieldWeight j t σ (badFieldFlat n j A ε m) y |
|
|
| lemma reducedBadDensity_le {n : ℕ} [NeZero n] {j t σ : ℝ} |
| (hj : 0 ≤ j) (ht : 0 ≤ t) (hσ : 0 < σ) (A ε m : ℝ) (y : Fin n → ℝ) : |
| reducedBadDensity j t σ A ε m y ≤ ENNReal.ofReal (scalarIntegrand j t σ y) := by |
| simpa only [reducedBadDensity,mul_one] using scalar_weighted_density_bound hj ht hσ y |
| (conditionalFieldWeight_le_one j t σ (badFieldFlat n j A ε m) y) |
|
|
| lemma empirical_of_not_forbidden {n : ℕ} [NeZero n] {R t0 T t σ : ℝ} {U : Set (ScalarTimeDomain R t0 T)} |
| {y : Fin n → ℝ} (hP : empiricalLaw y ∈ momentBall R) (ht : t∈Icc t0 T) |
| (hF : scalarEmpirical y t σ ∉ scalarRelativeForbidden U) : |
| (⟨empiricalLaw y,hP⟩,⟨t,ht⟩) ∈ U := by |
| by_contra h |
| exact hF ⟨(⟨empiricalLaw y,hP⟩,⟨t,ht⟩),h,rfl⟩ |
|
|
| lemma reduced_bad_integral_bound {n : ℕ} [NeZero n] {j t σ R t0 T R1 ν A ε m c : ℝ} |
| (hj : 0 ≤ j) (ht0 : 0 < t0) (ht : t∈Icc t0 T) (hσ : 0 < σ) (hR : 0 ≤ R) |
| (U : Set (ScalarTimeDomain R t0 T)) (hU : IsOpen U) |
| (hcond : ∀ x ∈ U, ∀ y : Fin n → ℝ, empiricalLaw y=(x.1:ProbabilityMeasure ℝ) → |
| (∑ i,if R1 < |y i| then y i^2 else 0) ≤ ν*(n:ℝ) → |
| (conditionalGoeLaw (j/(n:ℝ)) ((x.2:ℝ)+σ^2) (magnetization y) |
| (empiricalObservation j x.2 σ y)).real |
| {W | ((fun i k=>W (i,k)),y)∈badFieldSet n j A ε m} ≤ exp (-c*(n:ℝ))) : |
| (∫⁻ y,reducedBadDensity (n := n) j t σ A ε m y) ≤ |
| (∫⁻ y in {y : Fin n → ℝ | (n:ℝ)*R < ∑ i,y i^2},ENNReal.ofReal (scalarIntegrand j t σ y))+ |
| (∫⁻ y in {y : Fin n → ℝ | ∑ i,y i^2 ≤ (n:ℝ)*R} ∩ |
| {y | scalarEmpirical y t σ ∈ scalarRelativeForbidden U},ENNReal.ofReal (scalarIntegrand j t σ y))+ |
| (∫⁻ y in {y : Fin n → ℝ | (n:ℝ)*ν ≤ ∑ i,squareTail R1 (y i)},ENNReal.ofReal (scalarIntegrand j t σ y))+ |
| (∫⁻ y,ENNReal.ofReal (scalarIntegrand (ι := Fin n) j t σ y))*ENNReal.ofReal (exp (-c*(n:ℝ))) := by |
| let B : Set (Fin n → ℝ) := {y | ∑ i,y i^2 ≤ (n:ℝ)*R} |
| let F : Set (Fin n → ℝ) := {y | scalarEmpirical y t σ ∈ scalarRelativeForbidden U} |
| let D : Set (Fin n → ℝ) := {y | (n:ℝ)*ν ≤ ∑ i,squareTail R1 (y i)} |
| have hB : MeasurableSet B := (isClosed_le (by fun_prop) continuous_const).measurableSet |
| have hF : MeasurableSet F := ((isClosed_scalarRelativeForbidden hU).preimage |
| (continuous_empiricalLaw.prodMk continuous_const)).measurableSet |
| have hD : MeasurableSet D := measurableSet_le measurable_const (Finset.measurable_sum _ |
| (fun i _=>(measurable_squareTail R1).comp (measurable_pi_apply i))) |
| have hb := weighted_four_region_bound volume (reducedBadDensity (n := n) j t σ A ε m) |
| (fun y=>ENNReal.ofReal (scalarIntegrand j t σ y)) (measurable_scalarIntegrand j t σ).ennreal_ofReal |
| B F D hB hF hD (reducedBadDensity_le hj (ht0.le.trans ht.1) hσ A ε m) |
| (ENNReal.ofReal (exp (-c*(n:ℝ)))) (by |
| intro y hy |
| have hP := empirical_momentBall y hR (by simpa only [B,mem_ofPred_eq,Fintype.card_fin] using hy.1.1) |
| have hx := empirical_of_not_forbidden hP ht hy.1.2 |
| have htail : (∑ i,if R1 < |y i| then y i^2 else 0) ≤ ν*(n:ℝ) := by |
| have hh := le_of_lt (not_le.mp (show ¬ (n:ℝ)*ν ≤ ∑ i,squareTail R1 (y i) from hy.2)) |
| simpa only [squareTail,mul_comm] using hh |
| have hc' := hcond (⟨empiricalLaw y,hP⟩,⟨t,ht⟩) hx y rfl htail |
| exact scalar_weighted_density_bound hj (ht0.le.trans ht.1) hσ y |
| (conditionalFieldWeight_of_real_bound (badFieldFlat n j A ε m) y hc')) |
| have hBc : Bᶜ={y : Fin n → ℝ | (n:ℝ)*R < ∑ i,y i^2} := by ext y; change (¬ ∑ i,y i^2 ≤ (n:ℝ)*R) ↔ _; exact not_le |
| simpa only [hBc] using hb |
|
|
| theorem conditional_bad_integral_rate {j A t0 T : ℝ} (hj : 0 < j) (hj1 : j < 1) |
| (hA : 1 ≤ A) (hsub : sqrt j*A < 1) (ht0 : 0 < t0) (hT : t0 ≤ T) : |
| ∃ m ε σ0 a : ℝ, 0 < m ∧ 0 < ε ∧ 0 < σ0 ∧ 0 < a ∧ ∃ N : ℕ, |
| ∀ (n : ℕ) [NeZero n], N ≤ n → ∀ t ∈ Icc t0 T, ∀ σ ∈ Ioc 0 σ0, |
| (∫⁻ y,reducedBadDensity (n := n) j t σ A ε m y) ≤ ENNReal.ofReal (exp (-a*(n:ℝ))) := by |
| have hT0 : 0 ≤ T := ht0.le.trans hT |
| obtain ⟨R,hR,hTail⟩ := scalar_uniform_tail hj.le hT0 ht0 1 |
| obtain ⟨m,ν,σc,hm,hν,hσc,U,hU,hKU,hcond⟩ := conditional_field_stability hj hj1 hA hsub hR.le ht0 (T := T) |
| obtain ⟨R1,aD,haD,ND,hD⟩ := scalar_rate_iv hj hj1 ht0 hT hν |
| obtain ⟨ε,c,NC,hε,hc,_,hC⟩ := hcond (max R1 1) (le_max_right _ _) |
| obtain ⟨aF,haF,δ,hδ,NF,hF⟩ := scalar_relative_neighborhood_rate hj hj1 ht0 hT hU hKU |
| obtain ⟨NG,hG⟩ := scalar_unrestricted_rate hj hj1 hT0 ht0 (half_pos hc) |
| let b := min 1 (min aF (min aD (c/2))) |
| have hb : 0 < b := lt_min zero_lt_one (lt_min haF (lt_min haD (half_pos hc))) |
| obtain ⟨M,hM⟩ := absorb_finite_exponential_factor 4 (half_pos hb) |
| refine ⟨m,ε,min σc (min δ 1),b/2,hm,hε,lt_min hσc (lt_min hδ zero_lt_one),half_pos hb, |
| max NC (max ND (max NF (max NG M))),?_⟩ |
| intro n _ hn t ht σ hσ |
| have hn' : 0 < n := NeZero.pos n |
| have hNC : NC ≤ n := (le_max_left _ _).trans hn |
| have hND : ND ≤ n := (le_max_left ND _).trans ((le_max_right NC _).trans hn) |
| have hNF : NF ≤ n := (le_max_left NF _).trans ((le_max_right ND _).trans ((le_max_right NC _).trans hn)) |
| have hNG : NG ≤ n := (le_max_left NG M).trans ((le_max_right NF _).trans ((le_max_right ND _).trans ((le_max_right NC _).trans hn))) |
| have hMN : M ≤ n := (le_max_right NG M).trans ((le_max_right NF _).trans ((le_max_right ND _).trans ((le_max_right NC _).trans hn))) |
| have hσ1 : σ∈Icc 0 1 := ⟨hσ.1.le,(min_le_right _ _).trans' ((min_le_right _ _).trans' hσ.2)⟩ |
| have hσδ : σ∈Icc 0 (min δ 1) := ⟨hσ.1.le,(min_le_right _ _).trans' hσ.2⟩ |
| have hσc' : |σ| ≤ σc := by rw [abs_of_pos hσ.1]; exact (min_le_left _ _).trans' hσ.2 |
| let B : Set (Fin n → ℝ) := {y | ∑ i,y i^2 ≤ (n:ℝ)*R} |
| let F : Set (Fin n → ℝ) := {y | scalarEmpirical y t σ ∈ scalarRelativeForbidden U} |
| let D : Set (Fin n → ℝ) := {y | (n:ℝ)*ν ≤ ∑ i,squareTail (max R1 1) (y i)} |
| have hBm : MeasurableSet B := (isClosed_le (by fun_prop) continuous_const).measurableSet |
| have hFm : MeasurableSet F := ((isClosed_scalarRelativeForbidden hU).preimage |
| (continuous_empiricalLaw.prodMk continuous_const)).measurableSet |
| have hDm : MeasurableSet D := measurableSet_le measurable_const (Finset.measurable_sum _ |
| (fun i _=>(measurable_squareTail (max R1 1)).comp (measurable_pi_apply i))) |
| have hs (y : Fin n → ℝ) : t0 ≤ scalarS j (scalarEmpirical y t σ) := by |
| have hq := mul_nonneg hj.le (scalarQMoment_bounds (empiricalLaw y)).1 |
| change t0 ≤ t+σ^2+j*scalarQMoment (empiricalLaw y) |
| linarith [ht.1,sq_nonneg σ] |
| have htail := hTail n hn' t ⟨ht0.le.trans ht.1,ht.2⟩ σ hσ1 Bᶜ hBm.compl |
| (fun y _=>hs y) (fun y hy=>le_of_lt (not_le.mp hy)) |
| have hforb := hF n hNF hn' t ht σ hσδ (B∩F) (hBm.inter hFm) |
| (fun y hy=>empirical_momentBall y hR.le (by simpa only [B,mem_ofPred_eq,Fintype.card_fin] using hy.1)) (by |
| intro y hy x hx |
| obtain ⟨z,hz,hze⟩ := hy.2 |
| by_contra hh |
| push Not at hh |
| apply hz |
| have he : z=x := by |
| apply Prod.ext <;> apply Subtype.ext |
| · exact (congrArg Prod.fst hze).trans hh.1 |
| · exact (congrArg Prod.snd hze).trans hh.2 |
| simpa only [he] using hx) |
| have hdiag := hD (max R1 1) (le_max_left _ _) n hND hn' t ht σ hσ1 D hDm (fun _ hy=>hy) |
| have hglobal := hG n hNG hn' t ⟨ht0.le.trans ht.1,ht.2⟩ σ hσ1 univ MeasurableSet.univ (fun y _=>hs y) |
| simp only [Measure.restrict_univ] at hglobal |
| have hgood := mul_le_mul' hglobal (le_rfl (a := ENNReal.ofReal (exp (-c*(n:ℝ))))) |
| rw [← ENNReal.ofReal_mul (exp_pos _).le,← exp_add] at hgood |
| have he : c/2*(n:ℝ)+ -c*(n:ℝ)= -(c/2)*(n:ℝ) := by ring |
| rw [he] at hgood |
| have hcoeff {r : ℝ} (hr : b ≤ r) : ENNReal.ofReal (exp (-r*(n:ℝ))) ≤ ENNReal.ofReal (exp (-b*(n:ℝ))) := |
| ENNReal.ofReal_le_ofReal (exp_le_exp.mpr (by nlinarith [Nat.cast_nonneg (α := ℝ) n])) |
| have hb1 : b ≤ 1 := min_le_left _ _ |
| have hbF : b ≤ aF := (min_le_right _ _).trans (min_le_left _ _) |
| have hbD : b ≤ aD := (min_le_right _ _).trans ((min_le_right _ _).trans (min_le_left _ _)) |
| have hbC : b ≤ c/2 := (min_le_right _ _).trans ((min_le_right _ _).trans (min_le_right _ _)) |
| have hbound := reduced_bad_integral_bound hj.le ht0 ht hσ.1 hR.le U hU |
| (fun x hx y hP hy=>hC n hNC x hx y hP hy σ hσc') |
| have hBc : Bᶜ={y : Fin n → ℝ | (n:ℝ)*R < ∑ i,y i^2} := by ext y; change (¬ ∑ i,y i^2 ≤ (n:ℝ)*R) ↔ _; exact not_le |
| rw [← hBc] at hbound |
| have htotal := hbound.trans (add_le_add (add_le_add (add_le_add |
| (htail.trans (hcoeff hb1)) (hforb.trans (hcoeff hbF))) (hdiag.trans (hcoeff hbD))) |
| (hgood.trans (hcoeff hbC))) |
| have hadd : ENNReal.ofReal (exp (-b*(n:ℝ)))+ENNReal.ofReal (exp (-b*(n:ℝ)))+ |
| ENNReal.ofReal (exp (-b*(n:ℝ)))+ENNReal.ofReal (exp (-b*(n:ℝ)))= |
| ENNReal.ofReal (4*exp ((n:ℝ)*(-b))) := by |
| rw [← ENNReal.ofReal_add (exp_pos _).le (exp_pos _).le, |
| ← ENNReal.ofReal_add (by positivity) (exp_pos _).le, |
| ← ENNReal.ofReal_add (by positivity) (exp_pos _).le] |
| congr 1 |
| ring_nf |
| rw [hadd] at htotal |
| apply htotal.trans (ENNReal.ofReal_le_ofReal _) |
| convert hM n hMN (-b) using 1 |
| congr 1 |
| ring |
| end SKGap |
| end |
| end |
|
|
| end OAI |
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|
