| import OAI.Dynamics.StandardMap.FastBridge |
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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 |
| lemma strict_markov_real {X : Type*} [MeasurableSpace X] (μ : Measure X) |
| (H : X → ℝ) (hi : Integrable H μ) (h0 : ∀ x,0≤H x) (h : ℝ) (hh : 0<h) : |
| μ {x | h<H x} ≤ ENNReal.ofReal ((∫ x,H x ∂μ)/h) := by |
| have hb := meas_ge_le_lintegral_div hi.aestronglyMeasurable.aemeasurable.ennreal_ofReal |
| (ENNReal.ofReal_ne_zero_iff.mpr hh) ENNReal.ofReal_ne_top |
| rw [←ofReal_integral_eq_lintegral_ofReal hi (Eventually.of_forall h0)] at hb |
| have hs : {x | h<H x} ⊆ {x | ENNReal.ofReal h≤ENNReal.ofReal (H x)} := fun x hx => ENNReal.ofReal_le_ofReal hx.le |
| exact (measure_mono hs).trans (hb.trans_eq (ENNReal.ofReal_div_of_pos hh).symm) |
| lemma real_integral_bound_of_lintegral {X : Type*} [MeasurableSpace X] (μ : Measure X) |
| (W : X → ℝ) (hi : Integrable W μ) (h0 : ∀ x,0≤W x) (E : Set X) |
| (C : ℝ) (hC : 0≤C) (hb : (∫⁻ x in E,ENNReal.ofReal (W x) ∂μ)≤ENNReal.ofReal C) : |
| (∫ x in E,W x ∂μ) ≤ C := by |
| rw [←ofReal_integral_eq_lintegral_ofReal hi.restrict (Eventually.of_forall h0)] at hb |
| exact (ENNReal.ofReal_le_ofReal_iff hC).mp hb |
| noncomputable def bridgeRealConstant : ℝ := fastBridgeConstant.toReal |
| lemma bridgeRealConstant_nonneg : 0≤bridgeRealConstant := ENNReal.toReal_nonneg |
| lemma actual_fast_bridge_real (k : ℝ) (hc : BridgeScalarControl k) |
| (N r : ℕ) (hN : 100 ≤ N) (hr : (r:ℝ) ≤ (N:ℝ)/1000) |
| {Rw Rh : ℕ} (aw : Fin Rw → ℤ) (nw : Fin Rw → ℕ) (Fw : (Fin Rw → ℝ) → ℝ) |
| (ah : Fin Rh → ℤ) (nh : Fin Rh → ℕ) (Fh : (Fin Rh → ℝ) → ℝ) |
| (hnw : ∀ j, 0 < nw j) (hnh : ∀ j, 0 < nh j) (L : NNReal) |
| (hFw : LipschitzWith L Fw) (hFh : LipschitzWith L Fh) |
| (hlenw : ∀ j, nw j ≤ 2*r) (hlenh : ∀ j, nh j ≤ 2*r) |
| (hwinw : ∀ j l, 1 ≤ l → l ≤ nw j → (aw j+(l:ℤ)-1).natAbs ≤ r) |
| (hwinh : ∀ j l, 1 ≤ l → l ≤ nh j → (ah j+(l:ℤ)-1-(N:ℤ)).natAbs ≤ r) |
| (hW0 : ∀ z, 0 ≤ shortfallObservation k aw nw Fw z) |
| (hH0 : ∀ z, 0 ≤ shortfallObservation k ah nh Fh z) |
| (hlog : 1 ≤ Real.log (growthBase k)) (hLM : 16*Real.pi*L ≤ growthBase k) |
| (h : ℝ) (hh : 0<h) (hδ : growthBase k^(-(1/10:ℝ)*(N:ℝ)) < h/2) : |
| (∫ z in {z | torusFastBridge k z N} ∩ {z | h < shortfallObservation k ah nh Fh z}, |
| shortfallObservation k aw nw Fw z ∂area) ≤ |
| bridgeRealConstant*((∫ z,shortfallObservation k aw nw Fw z ∂area)+ |
| growthBase k^(-(1/10:ℝ)*(N:ℝ)))*((∫ z,shortfallObservation k ah nh Fh z ∂area)/(h/2)) := by |
| have hW := continuous_shortfall_observation k hc.nonneg aw nw Fw hFw.continuous |
| have hH := continuous_shortfall_observation k hc.nonneg ah nh Fh hFh.continuous |
| have hiW := hW.integrable_of_hasCompactSupport (μ := area) (HasCompactSupport.of_compactSpace _) |
| have hiH := hH.integrable_of_hasCompactSupport (μ := area) (HasCompactSupport.of_compactSpace _) |
| have hb := actual_fast_bridge_product k hc N r hN hr aw nw Fw ah nh Fh hnw hnh L hFw hFh |
| hlenw hlenh hwinw hwinh hW0 hlog hLM h hδ |
| have hd0 : 0≤growthBase k^(-(1/10:ℝ)*(N:ℝ)) := Real.rpow_nonneg (by linarith [growthBase_ge_four k hc.nonneg]) _ |
| have hiW0 := integral_nonneg_of_ae (μ := area) (Eventually.of_forall hW0) |
| have hiH0 := integral_nonneg_of_ae (μ := area) (Eventually.of_forall hH0) |
| apply real_integral_bound_of_lintegral area _ hiW hW0 _ _ |
| (mul_nonneg (mul_nonneg bridgeRealConstant_nonneg (add_nonneg hiW0 hd0)) (div_nonneg hiH0 (by linarith))) |
| have hm := strict_markov_real area _ hiH hH0 (h/2) (by linarith) |
| apply (hb.trans (mul_le_mul_of_nonneg_left hm bot_le)).trans_eq |
| change fastBridgeConstant*((∫⁻ z,ENNReal.ofReal (shortfallObservation k aw nw Fw z) ∂area)+ENNReal.ofReal (growthBase k^(-(1/10:ℝ)*(N:ℝ))))*ENNReal.ofReal ((∫ z,shortfallObservation k ah nh Fh z ∂area)/(h/2))=_ |
| rw [←ofReal_integral_eq_lintegral_ofReal (f := shortfallObservation k aw nw Fw) hiW (Eventually.of_forall hW0), |
| ←ENNReal.ofReal_add hiW0 hd0] |
| conv_lhs => rw [show fastBridgeConstant=ENNReal.ofReal bridgeRealConstant from (ENNReal.ofReal_toReal fastBridgeConstant_ne_top).symm] |
| rw [ |
| ←ENNReal.ofReal_mul bridgeRealConstant_nonneg,←ENNReal.ofReal_mul (mul_nonneg bridgeRealConstant_nonneg (add_nonneg hiW0 hd0))] |
| end StandardMapEntropy |
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| end OAI |
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