| import OAI.Dynamics.StandardMap.BandGeometry |
|
|
| namespace OAI |
|
|
| open MeasureTheory Set |
| open scoped ENNReal BigOperators |
|
|
| open Set Filter MeasureTheory Topology |
| open scoped ENNReal Classical |
| namespace StandardMapEntropy |
| noncomputable def bandCap (α:ℝ) (t:DyadicTime) (d:NonAffineArray) : ℝ := entropyCap α (arrayShortfall 0 t d.val) |
| lemma measurable_bandCap (α:ℝ) (t:DyadicTime) : Measurable (bandCap α t) := |
| ((entropyCap_contDiff α).continuous.comp ((continuous_arrayShortfall 0 t).comp continuous_subtype_val)).measurable |
| lemma bandCap_nonneg {μ:Measure NonAffineArray} {α:ℝ} (ha:0≤α) (ha1:α≤1) (t:DyadicTime) (ht:0<(t:ℝ)) : |
| 0≤ᵐ[bandLaw μ] bandCap α t := by |
| filter_upwards [ae_restrict_mem (measurableSet_lengthBand 0)] with d hd |
| exact (entropyCap_bounds ha ha1 (shortfall_unit_mem d.val hd.1 0 t (by simpa only [ZeroMemClass.coe_zero]))).1 |
| lemma bandCap_integrable (μ:Measure NonAffineArray) [IsFiniteMeasureOnCompacts μ] |
| (hs:∀ᵐd ∂μ,SlowShape (realArray d.val) (999/1000)) |
| (hT:∀r:DyadicTime,μ.map (nonaffineTranslation r)=μ) {α:ℝ} (ha:0≤α) (ha1:α≤1) |
| (t:DyadicTime) (ht:0<(t:ℝ)) : Integrable (bandCap α t) (bandLaw μ) := by |
| let E:Set NonAffineArray := {d | endpointLeft d.val∈Icc (-2) (t:ℝ)} |
| have hE : MeasurableSet E := (measurable_endpointLeft.comp measurable_subtype_coe) measurableSet_Icc |
| have hf : bandLaw μ E<∞ := bandLaw_endpoint_finite μ hs hT _ _ |
| have : IsFiniteMeasure ((bandLaw μ).restrict E) := ⟨by simpa only [Measure.restrict_apply_univ] using hf⟩ |
| have hi : Integrable (E.indicator (fun _ => (1:ℝ))) (bandLaw μ) := IntegrableOn.integrable_indicator (integrable_const (μ:=(bandLaw μ).restrict E) (1:ℝ)) hE |
| apply hi.mono' (measurable_bandCap α t).aestronglyMeasurable |
| filter_upwards [ae_restrict_mem (measurableSet_lengthBand 0),ae_restrict_of_ae hs] with d hd hsd |
| rw [Real.norm_eq_abs,bandCap,abs_of_nonneg ((entropyCap_bounds ha ha1 (shortfall_unit_mem d.val hd.1 0 t (by simpa only [ZeroMemClass.coe_zero]))).1)] |
| by_cases he:d∈E |
| · rw [indicator_of_mem he]; exact (entropyCap_bounds ha ha1 (shortfall_unit_mem d.val hd.1 0 t (by simpa only [ZeroMemClass.coe_zero]))).2 |
| · rw [indicator_of_notMem he] |
| exact le_of_eq (band_cap_support α d hd hsd t ht he) |
| lemma bandCap_one_fine {α:ℝ} (d:NonAffineArray) (hd:d∈lengthBand 0) |
| (hs:SlowShape (realArray d.val) (999/1000)) (he:endpointLeft d.val∈Ico (-1/2) 0) |
| (t:DyadicTime) (ht:0<(t:ℝ)) (ht1:(t:ℝ)≤1/2) : bandCap α t d=1 := by |
| obtain ⟨hab,hl,hr,hslow⟩ := twoRay_properties hd.1 (by norm_num) hs hd.2.1 |
| have hlen : 1≤endpointRight d.val-endpointLeft d.val := by simpa only [lengthBand,zpow_zero,coreLength] using hd.2.2.1 |
| have hb := hslow 0 t (by change (0:ℝ)∈Icc (endpointLeft d.val) (endpointRight d.val); constructor <;> linarith [he.1,he.2]) |
| (by constructor <;> linarith [he.1,he.2]) |
| rw [ZeroMemClass.coe_zero,sub_zero,abs_of_pos ht] at hb |
| have hr := (div_le_iff₀ ht).mpr hb |
| apply entropyCap_one |
| change 3/10000≤1-d.val.val 0 t/((t:ℝ)-(0:DyadicTime)) |
| rw [ZeroMemClass.coe_zero,sub_zero] |
| linarith |
| end StandardMapEntropy |
| |
| end OAI |
| |
