| import OAI.Dynamics.StandardMap.BridgeRealBound |
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| namespace OAI |
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| open MeasureTheory Set |
| open scoped ENNReal BigOperators |
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| open MeasureTheory Set Filter Metric |
| open scoped ENNReal Topology Classical |
| namespace StandardMapEntropy |
| noncomputable def arrayObservation {R : ℕ} (a b : Fin R → DyadicTime) |
| (F : (Fin R → ℝ) → ℝ) (d : DistanceArray) : ℝ := F (fun i => arrayShortfall (a i) (b i) d) |
| lemma continuous_arrayObservation {R : ℕ} (a b : Fin R → DyadicTime) |
| (F : (Fin R → ℝ) → ℝ) (hF : Continuous F) : Continuous (arrayObservation a b F) := |
| hF.comp (continuous_pi fun i => continuous_arrayShortfall (a i) (b i)) |
| lemma arrayObservation_sample {R : ℕ} (k : ℝ) (hk : 0≤ k) (n : ℕ) (hn : 0< n) |
| (a b : Fin R → DyadicTime) (A : Fin R → ℤ) (m : Fin R → ℕ) (hm : ∀i,0< m i) |
| (ha : ∀i,(n:ℝ)*(a i:ℝ)=(A i:ℝ)) (hb : ∀i,(n:ℝ)*(b i:ℝ)=(A i:ℝ)+(m i:ℝ)) |
| (F : (Fin R → ℝ) → ℝ) (z : Torus) : |
| arrayObservation a b F (sampleArray k hk z n hn)=shortfallObservation k A m F z := by |
| unfold arrayObservation shortfallObservation |
| congr 1 |
| funext i |
| exact shortfall_sample_aligned k hk z n hn (a i) (b i) (A i) (m i) (hm i) (ha i) (hb i) |
| lemma array_fast_sample (k : ℝ) (hk : 0≤ k) (z : Torus) (n : ℕ) (hn : 0< n) |
| (u v : DyadicTime) (a : ℤ) (N : ℕ) |
| (hu : (n:ℝ)*(u:ℝ)=(a:ℝ)) (hv : (n:ℝ)*(v:ℝ)=(a:ℝ)+(N:ℝ)) |
| (hf : (999/1000:ℝ)*((v:ℝ)-(u:ℝ))< (sampleArray k hk z n hn).val u v) : |
| torusFastBridge k (torusIter k a z) N := by |
| change _< rescaledDistance k z n u v at hf |
| rw [rescaledDistance_aligned k z n u v a (a+(N:ℤ)) hu (by push_cast; exact hv), |
| productDistance_forward] at hf |
| have hshift := torusSegmentTransfer_shift k z 0 a N |
| rw [zero_add] at hshift |
| unfold torusFastBridge |
| rw [hshift,Real.rpow_le_iff_le_log (by linarith [growthBase_ge_four k hk]) |
| (lt_of_lt_of_le zero_lt_one (torusSegmentTransfer_norm_bounds k hk z a N).1)] |
| have hnp : (0:ℝ)< n := by exact_mod_cast hn |
| have hg : 0< Real.log (growthBase k) := log_growthBase_pos k hk |
| have hl : (n:ℝ)*((v:ℝ)-(u:ℝ))=N := by linarith |
| have he := (lt_div_iff₀ hnp).mp hf |
| have he' := (lt_div_iff₀ hg).mp he |
| nlinarith |
| lemma integral_observable_sample {R : ℕ} (k : ℝ) (hk : 0≤ k) (n : ℕ) (hn : 0< n) |
| (a b : Fin R → DyadicTime) (A : Fin R → ℤ) (m : Fin R → ℕ) (hm : ∀i,0< m i) |
| (ha : ∀i,(n:ℝ)*(a i:ℝ)=(A i:ℝ)) (hb : ∀i,(n:ℝ)*(b i:ℝ)=(A i:ℝ)+(m i:ℝ)) |
| (F : (Fin R → ℝ) → ℝ) (hF : Continuous F) : |
| (∫ d,arrayObservation a b F d ∂sampleLaw k hk n hn)=∫ z,shortfallObservation k A m F z ∂area := by |
| rw [integral_sampleLaw k hk n hn _ (continuous_arrayObservation a b F hF)] |
| apply integral_congr_ae |
| exact Eventually.of_forall (arrayObservation_sample k hk n hn a b A m hm ha hb F) |
| lemma integral_supported_bound {X : Type*} [MeasurableSpace X] (μ : Measure X) |
| (F W : X → ℝ) (hF : Integrable F μ) (hW : Integrable W μ) (E : Set X) (hE : MeasurableSet E) |
| (D : ℝ) (_hD : 0≤ D) (_hW0 : ∀x,0≤ W x) |
| (hb : ∀x,x∈E → F x≤ D*W x) (hz : ∀x,x∉E → F x=0) : |
| (∫ x,F x ∂μ)≤ D*(∫ x in E,W x ∂μ) := by |
| rw [←integral_const_mul,←integral_indicator hE] |
| apply integral_mono hF ((hW.const_mul D).indicator hE) |
| intro x |
| by_cases hx : x∈E |
| · simpa only [indicator_of_mem hx] using hb x hx |
| · rw [hz x hx,indicator_of_notMem hx] |
| end StandardMapEntropy |
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| end OAI |
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