| import OAI.Dynamics.StandardMap.BandCoboundary |
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| namespace OAI |
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| open MeasureTheory Set |
| open scoped ENNReal BigOperators |
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| open Set Filter MeasureTheory Topology |
| open scoped ENNReal Classical |
| namespace StandardMapEntropy |
| lemma zoom_nat_one (n:ℕ) : dyadicZoom (n:ℤ) (dyadicInt 1)=dyadicInt ((2:ℕ)^n) := by |
| apply Subtype.ext |
| simp only [dyadicZoom_val,dyadicInt_val,Int.cast_one,mul_one,zpow_natCast,Nat.cast_ofNat,Int.cast_pow,Int.cast_ofNat] |
| lemma zoom_neg_one_le (n:ℕ) (hn:1≤n) : (dyadicZoom (-(n:ℤ)) (dyadicInt 1):ℝ)≤1/2 := by |
| rw [dyadicZoom_val,dyadicInt_val,Int.cast_one,mul_one] |
| have hh : (2:ℝ)^(-(n:ℤ))≤2^(-1:ℤ) := (zpow_right_strictMono₀ (by norm_num : (1:ℝ)<2)).monotone (by omega) |
| norm_num at hh ⊢; exact hh |
| lemma bandMoment_fine (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) (n:ℕ) (hn:1≤n) : |
| (bandIntensity μ (Ico 0 1)).toReal/2≤bandMoment μ α (-(n:ℤ)) := |
| band_fine_lower μ hs hT ha ha1 _ (zoom_one_pos _) (zoom_neg_one_le n hn) |
| lemma bandMoment_coarse (μ:Measure NonAffineArray) [IsFiniteMeasureOnCompacts μ] |
| (hs:∀ᵐd ∂μ,SlowShape (realArray d.val) (999/1000)) |
| (hT:∀r:DyadicTime,μ.map (nonaffineTranslation r)=μ) |
| (htree:∀ᵐd ∂μ,TreeArray d.val) (hcan:∀ᵐd ∂μ,NoFastCancellation (realArray d.val) (1/100000000)) |
| {α:ℝ} (ha:0≤α) (ha1:α≤1) (n:ℕ) (hn:1200000004/((2:ℝ)^n)≤1/10000) : |
| bandMoment μ α (n:ℤ)≤4*(α*1200000004)*(bandIntensity μ (Ico 0 1)).toReal := by |
| unfold bandMoment |
| rw [zoom_nat_one] |
| have hp:0<(2:ℕ)^n := pow_pos (by norm_num) _ |
| have hb:=band_coarse_upper μ hs hT htree hcan ha ha1 (2^n) hp (by exact_mod_cast hn) |
| refine hb.trans ?_ |
| have hh:(1:ℝ)≤2^n := one_le_pow₀ (by norm_num) |
| have haC:0≤α*1200000004 := mul_nonneg ha (by norm_num) |
| have hv : (α*1200000004/(2:ℝ)^n)*((2:ℝ)^n+3)≤4*(α*1200000004) := by |
| rw [div_mul_eq_mul_div,div_le_iff₀ (pow_pos (by norm_num) _)]; nlinarith |
| convert mul_le_mul_of_nonneg_right hv (ENNReal.toReal_nonneg) using 1 ; norm_cast |
| lemma bandMoment_order (μ:Measure NonAffineArray) [IsFiniteMeasureOnCompacts μ] |
| (hs:∀ᵐd ∂μ,SlowShape (realArray d.val) (999/1000)) |
| (hT:∀r:DyadicTime,μ.map (nonaffineTranslation r)=μ) |
| (htree:∀ᵐd ∂μ,TreeArray d.val) (hcan:∀ᵐd ∂μ,NoFastCancellation (realArray d.val) (1/100000000)) |
| {α:ℝ} (ha:0≤α) (ha1:α≤1) (hasmall:α*1200000004≤1/16) : |
| ∀ᶠn:ℕ in atTop, bandMoment μ α (n:ℤ)≤bandMoment μ α (-(n:ℤ)) := by |
| have hp:Tendsto (fun n:ℕ => (2:ℝ)^n) atTop atTop := tendsto_pow_atTop_atTop_of_one_lt (by norm_num) |
| have he:∀ᶠn:ℕ in atTop,1200000004/(2:ℝ)^n≤1/10000 := by |
| filter_upwards [hp.eventually (eventually_ge_atTop (1200000004*10000:ℝ))] with n hn |
| rw [div_le_iff₀ (pow_pos (by norm_num) _)]; linarith |
| filter_upwards [he,eventually_ge_atTop (1:ℕ)] with n hn hn1 |
| have h1:=bandMoment_coarse μ hs hT htree hcan ha ha1 n hn |
| have h2:=bandMoment_fine μ hs hT ha ha1 n hn1 |
| refine le_trans h1 (le_trans ?_ h2) |
| have hI:=ENNReal.toReal_nonneg (a:=bandIntensity μ (Ico 0 1)) |
| nlinarith [mul_nonneg (by linarith : 0≤1/2-4*(α*1200000004)) hI] |
| end StandardMapEntropy |
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| end OAI |
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