File size: 3,486 Bytes
92416a2 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 | import OAI.Dynamics.StandardMap.BridgeRealBound
namespace OAI
open MeasureTheory Set
open scoped ENNReal BigOperators
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
end OAI
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